Class 11 - Maths - Permutations Combinations

Exercise 7.1

Question 1:

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that            (i) repetition of the digits is allowed?

(ii) repetition of the digits is not allowed?

Answer:

(i) There will be as many ways as there are ways of filling 3 vacant places in succession by the

given five digits. In this case, repetition of digits is allowed.

Therefore, the units place can be filled in by any of the given five digits.

Similarly, tens and hundreds digits can be filled in by any of the given five digits.

Thus, by the multiplication principle, the number of ways in which three-digit numbers can be

formed from the given digits is 5 * 5 * 5 = 125

(ii) In this case, repetition of digits is not allowed. Here, if units place is filled in first, then it

can be filled by any of the given five digits.

Therefore, the number of ways of filling the units place of the three-digit number is 5.

Then, the tens place can be filled with any of the remaining four digits and the hundreds place

can be filled with any of the remaining three digits.

Thus, by the multiplication principle, the number of ways in which three-digit numbers can be

formed without repeating the given digits is 5 * 4 * 3 = 60

Question 2:

How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Answer:

There will be as many ways as there are ways of filling 3 vacant places in succession by the

given six digits.

In this case, the units place can be filled by 2 or 4 or 6 only i.e., the units place can be filled in 3

ways.

The tens place can be filled by any of the 6 digits in 6 different ways and also the hundreds

place can be filled by any of the 6 digits in 6 different ways, as the digits can be repeated.

Therefore, by multiplication principle, the required number of three digit even numbers is

3 * 6 * 6 = 108

Question 3:

How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Answer:

There are as many codes as there are ways of filling 4 vacant places in succession by the first

10 letters of the English alphabet, keeping in mind that the repetition of letters is not allowed.

The first place can be filled in 10 different ways by any of the first 10 letters of the English

alphabet following which, the second place can be filled in by any of the remaining letters in 9

different ways. The third place can be filled in by any of the remaining 8 letters in 8 different

ways and the fourth place can be filled in by any of the remaining 7 letters in 7 different ways.

Therefore, by multiplication principle, the required numbers of ways in which 4 vacant places

can be filled is 10 * 9 * 8 * 7 = 5040

Hence, 5040 four-letter codes can be formed using the first 10 letters of the English alphabet,

if no letter is repeated.

Question 4:

How many 5–digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?

Answer:

It is given that the 5-digit telephone numbers always start with 67.

Therefore, there will be as many phone numbers as there are ways of filling 3 vacant places

6, 7, _ , _, _ by the digits 0 – 9, keeping in mind that the digits cannot be repeated.

The units place can be filled by any of the digits from 0 – 9, except digits 6 and 7.

Therefore, the units place can be filled in 8 different ways following which, the tens place can

be filled in by any of the remaining 7 digits in 7 different ways, and the hundreds place can be

filled in by any of the remaining 6 digits in 6 different ways.

Therefore, by multiplication principle, the required number of ways in which 5-digit telephone

numbers can be constructed is 8 * 7 * 6 = 336

Question 5:

A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

Answer:

When a coin is tossed once, the number of outcomes is 2 (Head and tail) i.e., in each throw,

the number of ways of showing a different face is 2.

Thus, by multiplication principle, the required number of possible outcomes is 2 * 2 * 2 = 8

Question 6:

Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

Answer:

Each signal requires the use of 2 flags.

There will be as many flags as there are ways of filling in 2 vacant places in succession by the

given 5 flags of different colours.

The upper vacant place can be filled in 5 different ways by any one of the 5 flags following

which, the lower vacant place can be filled in 4 different ways by any one of the remaining 4

different flags.

Thus, by multiplication principle, the number of different signals that can be generated is

5 * 4 = 20.

Exercise 7.2

Question 1:

Evaluate (i) 8!                                         (ii) 4! – 3!

Answer:

We know that

n! = n(n - 1)(n - 2)(n - 3)……………..3.2.1

or

n! = 1.2.3………..(n - 2)(n - 1)n

(i) 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320

(ii) 4! = 4 * 3 * 2 * 1 = 24

3! = 3 * 2 * 1 = 6

Now, 4! – 3! = 24 – 6 = 18

Question 2:

Is 3! + 4! = 7!?

Answer:

We know that

n! = n(n - 1)(n - 2)(n - 3)……………..3.2.1

or

n! = 1.2.3………..(n - 2)(n - 1)n

3! = 1 * 2 * 3 = 6

4! = 1 * 2 * 3 * 4 = 24

Now, 3! + 4! = 6 + 24 = 30

7! = 1 * 2 * 3 * 4 * 5 * 6 * 7 = 5040     So, 3! + 4! ≠ 7!

Question 3:

Compute 8!/(6! * 2!)

Answer:

8!/(6! * 2!) = (8 * 7 * 6!)/(6! * 2 * 1) = 56/2 = 28

Question 4:

If 1/6! + 1/7! = x/8!, find x.

Answer:

Given, 1/6! + 1/7! = x/8!

=> 1/6! + 1/(7 * 6!) = x/(8 * 7 * 6!)

=> (1/6!)(1 + 1/7) = x/(8 * 7 * 6!)

=> 1 + 1/7 = x/56

=> 8/7 = x/56

=> x = 8 * 56

=> x = (8 * 56)/8

=> x = 8 * 8

=> x = 64

So, the value of x is 64.

Question 5:

Evaluate n!/(n - r)!, when

(i) n = 6, r = 2                                  (ii) n = 9, r = 5

Answer:

(i) Given, n = 6, r = 2

Now, n!/(n - r)! = 6!/(6 - 2)! = 6!/4! = (6 * 5 * 4!)/4! = 6 * 5 = 30

(ii) Given, n = 9, r = 5

Now, n!/(n - r)! = 9!/(9 - 5)! = 9!/4!

= (9 * 8 * 7 * 6 * 5 * 4!)/4!

= 9 * 8 * 7 * 6 * 5

= 15120

Exercise 7.3

Question 1:

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

Answer:

3-digit numbers have to be formed using the digits 1 to 9.

Here, the order of the digits matters.

Therefore, there will be as many 3-digit numbers as there are permutations of 9 different

digits taken 3 at a time.

Therefore, required number of 3-digit numbers = 9P3 = 9!/(9 – 3)!

= 9!/6!

= (9 * 8 * 7 * 6!)/6!

= 9 * 8 * 7

= 504

Question 2:

How many 4-digit numbers are there with no digit repeated?

Answer:

The thousands place of the 4-digit number is to be filled with any of the digits from 1 to 9 as

the digit 0 cannot be included. Therefore, the number of ways in which thousands place can

be filled is 9.

The hundreds, tens, and units place can be filled by any of the digits from 0 to 9. However, the

digits cannot be repeated in the 4-digit numbers and thousands place is already occupied with

a digit. The hundreds, tens, and units place is to be filled by the remaining 9 digits.

Therefore, there will be as many such 3-digit numbers as there are permutations of 9 different

digits taken 3 at a time.

Number of such 3-digit numbers = 9P3 = 9!/(9 – 3)!

= 9!/6!

= (9 * 8 * 7 * 6!)/6!

= 9 * 8 * 7

= 504

Thus, by multiplication principle, the required number of 4-digit numbers is 9 * 504 = 4536

Question 3:

How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?

Answer:

3-digit even numbers are to be formed using the given six digits, 1, 2, 3, 4, 6, and 7 without

repeating the digits.

Then, units digits can be filled in 3 ways by any of the digits, 2, 4, or 6.

Since the digits cannot be repeated in the 3-digit numbers and units place is already occupied

with a digit (which is even), the hundreds and tens place is to be filled by the remaining 5

digits.

Therefore, the number of ways in which hundreds and tens place can be filled with the

remaining 5 digits is the permutation of 5 different digits taken 2 at a time.

Number of ways of filling hundreds and tens place = 5P2 = 5!/(5 – 2)!

= 5!/3!

= (5 * 4 * 3!)/3!

= 5 * 4 =20

Thus, by multiplication principle, the required number of 3-digit numbers is 3 * 20 = 60

Question 4:

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Answer:

4-digit numbers are to be formed using the digits, 1, 2, 3, 4, and 5.

There will be as many 4-digit numbers as there are permutations of 5 different digits taken 4

at a time.

Therefore, required number of 4 digit numbers = 5P4 = 5!/(5 – 4)!

= 5!/1!

= 5 * 4 * 3 * 2 * 1

= 120

Among the 4-digit numbers formed by using the digits, 1, 2, 3, 4, 5, even numbers end with

either 2 or 4.

The number of ways in which units place is filled with digits is 2.

Since the digits are not repeated and the units place is already occupied with a digit (which

is even), the remaining places are to be filled by the remaining 4 digits. Therefore, the number

of ways in which the remaining places can be filled is the permutation of 4 different digits

taken 3 at a time.

Number of ways of filling the remaining places = 4P3 = 4!/(4 – 3)!

= 4!/1!

= 4 * 3 * 2 * 1

= 24

Thus, by multiplication principle, the required number of even numbers is 24 * 2 = 48

Question 5:

From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?

Answer:

From a committee of 8 persons, a chairman and a vice chairman are to be chosen in such

a way that one person cannot hold more than one position.

Here, the number of ways of choosing a chairman and a vice chairman is the permutation of 8

different objects taken 2 at a time.

Thus, required number of ways = 8P2 = 8!/(8 – 2)!

= 8!/6!

= (8 * 7 * 6!)/6!

= 8 * 7

= 56

Question 6:

Find n if n-1P3 : nP4 = 1 : 9

Answer:

Given, n-1P3 : nP4 = 1 : 9

=> n-1P3 / nP4 = 1/9

=> {(n - 1)!/(n – 1 - 3)!}/ {n!/(n – 4)!} = 1/9

=> {(n - 1)!/(n – 4)!}/ {n!/(n – 4)!} = 1/9

=> (n - 1)!/n! = 1/9

=> (n - 1)!/{n * (n - 1)! = 1/9

=> 1/n = 1/9

=> n = 9

Question 7:

Find r if (i) 5Pr = 2 * 6Pr-1                                (ii) 5Pr = 6Pr-1

Answer:

(i) 5Pr = 2 * 6Pr-1

=> 5!/(5 - r)! = 2 * 6!/(6 – r + 1)!

=> 5!/(5 - r)! = 2 * 6!/(7 – r)!

=> 5!/(5 - r)! = 2 * (6 * 5!)/{(7 – r)(6 – r)(5 – r)!}

=> 1 = (2 * 6)/ {(7 – r)(6 – r)}

=> (7 – r)(6 – r) = 12

=> 42 – 6r – 7r + r2 = 12

=> r2 – 13r + 30 = 0

=> r2 – 3r – 10r + 30 = 0

=> r(r - 3) – 10(r - 3) = 0

=> (r - 3)(r - 10) = 0

=> r = 3, 10

It is known that

nPr = n!/(n - r)!, where 0 ≤ r ≤ n

So, 0 ≤ r ≤ 5

Hence, r ≠ 10

Therefore, r = 3

(ii) 5Pr = 6Pr-1

=> 5!/(5 - r)! = 6!/(6 – r + 1)!

=> 5!/(5 - r)! = 6!/(7 – r)!

=> 5!/(5 - r)! = (6 * 5!)/{(7 – r)(6 – r)(5 – r)!}

=> 1 = 6/ {(7 – r)(6 – r)}

=> (7 – r)(6 – r) = 6

=> 42 – 6r – 7r + r2 = 6

=> r2 – 13r + 36 = 0

=> r2 – 4r – 9r + 36 = 0

=> r(r - 4) – 9(r - 4) = 0

=> (r - 4)(r - 9) = 0

=> r = 4, 9

It is known that

nPr = n!/(n - r)!, where 0 ≤ r ≤ n

So, 0 ≤ r ≤ 5

Hence, r ≠ 10

Therefore, r = 3

Question 8:

How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?

Answer:

There are 8 different letters in the word EQUATION.

Therefore, the number of words that can be formed using all the letters of the word

EQUATION, using each letter exactly once, is the number of permutations of 8 different

objects taken 8 at a time, which is 8P8 = 8!

Thus, required number of words that can be formed = 8! = 40320

Question 9:

How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if

(i) 4 letters are used at a time,

(ii) all letters are used at a time,

(iii) all letters are used but first letter is a vowel?

Answer:

There are 6 different letters in the word MONDAY.

(i) Number of 4-letter words that can be formed from the letters of the word MONDAY,

without repetition of letters, is the number of permutations of 6 different objects taken 4 at a

time, which is 6P4

Thus, required number of words that can be formed using 4 letters at a time is

6P4 = 6!/(6 - 4)! = 6!/2! = (6 * 5 * 4 * 3 *2 *1)/(2 * 1) = 6 * 5 * 4 * 3 = 360

(ii) Number of words that can be formed by using all the letters of the word MONDAY at a time

is the number of permutations of 6 different objects taken 6 at a time, which is 6P6 = 6!

Thus, required number of words that can be formed when all letters are used at a time

= 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720

(iii) In the given word, there are 2 different vowels, which have to occupy the rightmost place

of the words formed. This can be done only in 2 ways.

Since the letters cannot be repeated and the rightmost place is already occupied with a letter

(which is a vowel), the remaining five places are to be filled by the remaining 5 letters. This can

be done in 5! ways.

Thus, in this case, required number of words that can be formed is 5! * 2 = 120 * 2 = 240

Question 10:

In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?

Answer:

In the given word MISSISSIPPI, I appears 4 times, S appears 4 times, P appears 2 times, and M

appears just once.

Therefore, number of distinct permutations of the letters in the given word

= 11!/(4! * 4! * 2!)

= (11 * 10 * 9 * 8 * 7 * 6 * 5 *4!)/(4! * 4 * 3 * 2 * 1 * 2 * 1)

= 11 * 10 * 9 * 8 * 7 * 6 * 5/(4 * 3 * 2 * 1 * 2 * 1)

= 34650

 I I I I

There are 4 Is in the given word. When they occur together, they are treated as a single object

for the time being. This single object together with the remaining 7 objects will

account for 8 objects.

These 8 objects in which there are 4 Ss and 2 Ps can be arranged in 8!/(4! * 2!) ways i.e.,

840 ways.

Number of arrangements where all Is occur together = 840

Thus, number of distinct permutations of the letters in MISSISSIPPI in which four Is do not

come together = 34650 – 840 = 33810

Question 11:

In how many ways can the letters of the word PERMUTATIONS be arranged if the

(i) words start with P and end with S,

(ii)vowels are all together,

(iii) there are always 4 letters between P and S?

Answer:

In the word PERMUTATIONS, there are 2 Ts and all the other letters appear only once.

(i) If P and S are fixed at the extreme ends (P at the left end and S at the right end), then 10

letters are left.

Hence, in this case, required number of arrangements = 10!/2! = 1814400

(ii) There are 5 vowels in the given word, each appearing only once.

Since they have to always occur together, they are treated as a single object for the time

being. This single object together with the remaining 7 objects will account for 8 objects.

These 8 objects in which there are 2 Ts can be arranged in 8!/2! ways.

Corresponding to each of these arrangements, the 5 different vowels can be arranged in 5!

ways.

Therefore, by multiplication principle, required number of arrangements in this case

= 8!/2! * 5 = 2419200

(iii) The letters have to be arranged in such a way that there are always 4 letters between P

and S.

Therefore, in a way, the places of P and S are fixed. The remaining 10 letters in which there are

2 Ts can be arranged in 10!/2! ways.

Also, the letters P and S can be placed such that there are 4 letters between them in 2 * 7 = 14

ways.

Therefore, by multiplication principle, required number of arrangements in this case

= 10!/2! * 14 = 25401600

Exercise 7.4

Question 1:

If nC8 = nC2, find nC2

Answer:

We know that nCa = nCb

=> n = a + b or a = b

Now,

nC8 = nC2

=> n = 8 + 2 = 10

So, 10C2 = 10!/{2! * (10 - 2)!}

= 10!/(2! * 8!)

= (10 * 9 * 8!)/(2 * 1 * 8!)

= 90/2

= 45

Question 2:

(i) 2nC3 : nC3 = 12 : 1                                            (ii) 2nC3 : nC3 = 11 : 1

Answer:

(i) 2nC3 : nC3 = 12 : 1

=> 2nC3 / nC3 = 12/1

=> [(2n)!/{3! * (2n - 3)}]/[(n)!/{3! * (n - 3)}] = 12/1

=> [{(2n) *(2n - 1) *(2n - 2) * (2n - 3)!}]/[{3! * (2n - 3)}/[{n * (n -1) *(n -2) * (n - 3)!}/{3! * (n –

3)}] = 12/1

=> {2(2n - 1)(2n - 2)}/{(n -1)(n - 2)} = 12

=> {4(2n - 1)(n - 1)}/{(n -1)(n - 2)} = 12

=> (2n - 1)/(n - 2) = 3

=> 2n - 1 = 3(n - 2)

=> 2n – 1 = 3n – 6

=> 3n – 2n = 6 – 1

=> n = 5

(ii) 2nC3 : nC3 = 11 : 1

=> 2nC3 / nC3 = 11/1

=> [(2n)!/{3! * (2n - 3)}]/[(n)!/{3! * (n - 3)}] = 11/1

=> [{(2n) *(2n - 1) *(2n - 2) * (2n - 3)!}]/[{3! * (2n - 3)}/[{n * (n -1) *(n -2) * (n - 3)!}/{3! * (n –

3)}] = 11

=> {2(2n - 1)(2n - 2)}/{(n -1)(n - 2)} = 11

=> {4(2n - 1)(n - 1)}/{(n -1)(n - 2)} = 11

=> 4(2n - 1)/(n - 2) = 11

=> 8n - 4 = 11(n - 2)

=> 8n – 4 = 11n – 22

=> 11n – 8n = 22 – 4

=> 3n = 18

=> n = 18/3

=> n = 6

Question 3:

How many chords can be drawn through 21 points on a circle?

Answer:

For drawing one chord on a circle, only 2 points are required.

To know the number of chords that can be drawn through the given 21 points on a circle, the

number of combinations have to be counted.

Therefore, there will be as many chords as there are combinations of 21 points taken 2 at a

time.

Thus, required number of chords = 21C2 = (21)!/{2! * (21 - 2)!}

= (21)!/{2! * 19!}

= (21 * 20 * 19!)/{2 * 1 * 19!}

= 420/2

= 210

Question 4:

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Answer:

A team of 3 boys and 3 girls is to be selected from 5 boys and 4 girls.

3 boys can be selected from 5 boys in 5C3 ways.

3 girls can be selected from 4 girls in 4C3 ways.

Therefore, by multiplication principle, number of ways in which a team of 3 boys and 3 girls

can be selected = 5C3 * 4C3

= {5!/(3! * 2!)} * {4!/(3! * 1!)}

= {(5 * 4 * 3!)/(3! * 2)} * {(4 * 3)!/3!}

= (20/2) * 4  = 10 * 4  = 40

Question 5:

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Answer:

There are a total of 6 red balls, 5 white balls, and 5 blue balls.

9 balls have to be selected in such a way that each selection consists of 3 balls of each colour.

Here,

3 balls can be selected from 6 red balls in 6C3 ways.

3 balls can be selected from 5 white balls in 5C3 ways.

3 balls can be selected from 5 blue balls in 5C3 ways.

Thus, by multiplication principle, required number of ways of selecting 9 balls girls can be

selected = 6C3 * 5C3 * 5C3

= {6!/(3! * 3!)} * {5!/(3! * 2!)} * {5!/(3! * 2!)}

= {6 * 5 * 4 * 3!)/(3! * 3 * 2)} * {(5 * 4 * 3!)/(3! * 2)} * {(5 * 4 * 3!)/(3! * 2)}

= 20 * 10 * 10

= 2000

Question 6:

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Answer:

In a deck of 52 cards, there are 4 aces.

A combination of 5 cards have to be made in which there is exactly one ace.

Then, one ace can be selected in 4C1 ways and the remaining 4 cards can be selected out of the

48 cards in 48C4 ways.

Thus, by multiplication principle, required number of 5 card combinations

= 48C4 * 4C1

= [(48)!/{4! * 44!}] * [(4)!/{1! * 3!}]

= [(48 * 47 * 46 * 45 * 44!)/(4! * 44!)] * [4!/ * 3!]

= [(48 * 47 * 46 * 45)/(4 * 3 * 2 * 1)] * [(4 * 3!)/ * 3!]

= (48 * 47 * 46 * 45)/(4 * 3 * 2 * 1)] * 4

= (48 * 47 * 46 * 45)/(3 * 2 * 1)]

= 8 * 47 * 46 * 45

= 778320

Question 7:

In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Answer:

Out of 17 players, 5 players are bowlers.

A cricket team of 11 players is to be selected in such a way that there are exactly 4 bowlers.

4 bowlers can be selected in 5C4 ways and the remaining 7 players can be selected out of the

12 players in 12C7 ways.

Thus, by multiplication principle, required number of ways of selecting cricket team

= 5C4 * 12C7

= 5!/(4! * 1!) * 12!/(7! * 5!)

= 5 * (12 * 11 * 10 * 9 * 8)/(5 * 4 * 3 * 2 * 1)

= (12 * 11 * 10 * 9 * 8)/(4 * 3 * 2 * 1)

= 11 * 10 * 9 * 4

= 3960

Question 8:

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Answer:

There are 5 black and 6 red balls in the bag.

2 black balls can be selected out of 5 black balls in 5C2 ways and 3 red balls can be selected

out of 6 red balls in 6C3 ways.

Thus, by multiplication principle, required number of ways of selecting 2 black and 3 red balls

= 5C2 * 6C3

= 5!/(2! * 3!) * 6!/(3! * 3!)

= {(5 * 4)/2} * {(6 * 5 * 4)/(3 * 2 * 1)}

=10 * 20

= 200

Question 9:

In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

Answer:

There are 9 courses available out of which, 2 specific courses are compulsory for every

student.

Therefore, every student has to choose 3 courses out of the remaining 7 courses.

This can be chosen in 7C3 ways.

Thus, required number of ways of choosing the programme = 7C3 = 7!/(3! * 4!)

= (7 * 6 * 5 * 4!)/(3 * 2 * 1 * 4!)

= (7 * 6 * 5)/6 = 35

Miscellaneous Exercise on Chapter 7

Question 1:

How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?

Answer:

In the word DAUGHTER, there are 3 vowels namely, A, U, and E, and 5 consonants namely, D,

G, H, T, and R.

Number of ways of selecting 2 vowels out of 3 vowels = 3C2 = 3

Number of ways of selecting 3 consonants out of 5 consonants = 5C3 = 10

Therefore, number of combinations of 2 vowels and 3 consonants = 3 * 10 = 30

Each of these 30 combinations of 2 vowels and 3 consonants can be arranged among

themselves in 5! ways.

Hence, required number of different words = 30 * 5! = 30 * 120 = 3600

Question 2:

How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

Answer:

In the word EQUATION, there are 5 vowels, namely, A, E, I, O, and U, and 3 consonants,

namely, Q, T, and N.

Since all the vowels and consonants have to occur together, both (AEIOU) and (QTN) can be

assumed as single objects. Then, the permutations of these 2 objects taken all at a time are

counted.

This number would be 2P2 = 2!

Corresponding to each of these permutations, there are 5! permutations of the five vowels

taken all at a time and 3! permutations of the 3 consonants taken all at a time.

Hence, by multiplication principle, required number of words = 2! * 5! * 3!

= 2 * 120 * 6

= 1440

Question 3:

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:                                                                                                   (i) exactly 3 girls?                          (ii) at least 3 girls?                          (iii) at most 3 girls?

Answer:

Given number of boys = 9

Number of girls = 4

Now, A committee of 7 has to be formed from 9 boys and 4 girls.

1. If in committee consist of exactly 3 girls:

4C3 * 9C4

= {4! / (3! * 1!)} * {9! / (4! * 5!)}

= {(4*3!) /3!} * {(9*8*7*6*5!) / (4! * 5!)}

= 4 * {(9*8*7*6*) / 4!}

= {4 * (9*8*7*6*)} / (4*3*2*1)

= 9*8*7

= 504

1. If committee consists of at least 3 girls:

4C3 * 9C4 + 4C4 * 9C3

= [{4! / (3! * 1!)} * {9! / (4! * 5!)}] + 9C3

= [{(4*3!) /3!} * {(9*8*7*6*5!) / (4! * 5!)}] + 9! /(3! * 6!)

= [4 * {(9*8*7*6*) / 4!}] + (9*8*7*6!)/(3! * 6!)

= [{4 * (9*8*7*6*)} / (4*3*2*1)] + (9*8*7)/3!

= (9*8*7) + (9*8*7)/(3*2*1)

= 504 + (504/6)

= 504 + 84

= 588

1. If committee consists of at most 3 girls:

= 4C0 * 9C7 + 4C1 * 9C6 + 4C2 * 9C5 + 4C3 * 9C4

= 36 + 336 + 756 + 504

= 1632

Question 4:

If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary,

how many words are there in this list before the first word starting with E?

Answer:

In the given word EXAMINATION, there are 11 letters out of which, A, I, and N appear 2 times

and all the other letters appear only once.

The words that will be listed before the words starting with E in a dictionary will be the words

that start with A only.

Therefore, to get the number of words starting with A, the letter A is fixed at the extreme left

position, and then the remaining 10 letters taken all at a time are rearranged.

Since there are 2 Is and 2 Ns in the remaining 10 letters,

Number of words starting with A =10!/(2! * 2!) = 907200

Thus, the required numbers of words is 907200.

Question 5:

How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?

Answer:

A number is divisible by 10 if the unit digit of the number is 0.

Given digits are: 0, 1, 3, 5, 7, 9

Now we fix digit 0 at unit place of the number.

Remaining 5 digits can be arranged in 5! ways

So, total 6-digit numbers which are divisible by 10 = 5! = 120

Question 6:

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?

Answer:

Number of vowels = 5

Number of consonants = 21

Now we have to form words with 2 different vowels and 2 different consonants.

Number of words = 5C2 * 21C2 * 4!    (4! is because we can arrange the letters in 4! ways)

= (5*4)/2 * (21*20)/2 * 24    (4! = 24)

= 20/2 * 420/2 * 24

= 10*210*24

= 50400

So, total number of words = 50400

Question 7:

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively.

A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Answer:

Given, total number of questions = 12

Part 1 contains 5 questions and part 2 contain 7 questions.

A student has to select 8 questions all in which there is at least 3 questions from each part.

Now different possibilities can occur.

Part1 (5 questions)            Part2 (7 questions)

3                                    5

4                                    4

5                                    3

Now total number of ways is

(5C3 * 7C5) + (5C4 * 7C4) + (5C5 * 7C3)

= (5C2 * 7C2) + (5C1 * 7C3) + (5C0 * 7C3)

= {(5 * 4 * 7 * 6)/(2 * 2)} + {(5 * 7 * 6 * 5)/(3 * 2)} + {(7 * 6 * 5)/(3 * 2)}

= 210 + 175 + 35

= 420

So, the total number of ways a student can select the questions is 420

Question 8:

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

Answer:

From a deck of 52 cards, 5-card combinations have to be made in such a way that in each

selection of 5 cards, there is exactly one king.

In a deck of 52 cards, there are 4 kings.

1 king can be selected out of 4 kings in 4C1 ways.

4 cards out of the remaining 48 cards can be selected in 48C4 ways.

Thus, the required number of 5-card combinations = 4C1 * 48C4

Question 9:

It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

Answer:

4 men and 4 women are to be seated in a row such that the women occupy the even places.

The 5 men can be seated in 5! ways. For each arrangement, the 4 women can be seated only

at the cross marked places (so that women occupy the even places).

M * M * M * M * M

Therefore, the women can be seated in 4! ways.

Thus, possible number of arrangements = 4! * 5! = 24 * 120 = 2880

Question 10:

From a class of 25 students, 10 are to be chosen for an excursion party.

There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Answer :

From the class of 25 students, 10 are to be chosen for an excursion party.

Since there are 3 students who decide that either all of them will join or none of them will

join, there are two cases.

Case I: All the three students join.

Then, the remaining 7 students can be chosen from the remaining 22 students in 22C7 ways.

Case II: None of the three students join.

Then, 10 students can be chosen from the remaining 22 students in 22C10 ways.

Thus, required number of ways of choosing the excursion party = 22C7 + 22C10

Question 11:

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together?

Answer:

In the given word ASSASSINATION, the letter A appears 3 times, S appears 4 times, I appears 2

times, N appears 2 times, and all the other letters appear only once. Since all the words have

to be arranged in such a way that all the Ss are together, SSSS is treated as a single object for

the time being. This single object together with the remaining 9 objects will account for 10

objects.

These 10 objects in which there are 3 As, 2 Is, and 2 Ns can be arranged in 10!/(3! * 2! * 2!)

ways.

Thus, required number of ways of arranging the letters of the given word

= 10!/(3! * 2! * 2!)

= (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3!/(3! * 2 * 2)

= 10 * 9 * 8 * 7 * 6 * 5

= 151200

.