12
Q:

# In how many ways can 5 different toys be packed in 3 identical boxes such that no box is empty, if any of the boxes may hold all of the toys ?

 A) 36 B) 25 C) 24 D) 72

Explanation:

The toys are different; The boxes are identical

If none of the boxes is to remain empty, then we can pack the toys in one of the following ways

a. 2, 2, 1

b. 3, 1, 1

Case a. Number of ways of achieving the first option 2 - 2 - 1

Two toys out of the 5 can be selected in $5{C}_{2}$ ways. Another 2 out of the remaining 3 can be selected in $3{C}_{2}$ ways and the last toy can be selected in $1{C}_{1}$ way.

However, as the boxes are identical, the two different ways of selecting which box holds the first two toys and which one holds the second set of two toys will look the same. Hence, we need to divide the result by 2

Therefore, total number of ways of achieving the 2 - 2 - 1 option is ways $5{C}_{2}*3{C}_{2}$= 15 ways

Case b. Number of ways of achieving the second option 3 - 1 - 1

Three toys out of the 5 can be selected in $5{C}_{3}$ ways. As the boxes are identical, the remaining two toys can go into the two identical looking boxes in only one way.

Therefore, total number of ways of getting the 3 - 1 - 1 option is $5{C}_{3}$ = 10 = 10 ways.

Total ways in which the 5 toys can be packed in 3 identical boxes

= number of ways of achieving Case a + number of ways of achieving Case b= 15 + 10 = 25 ways.

Q:

From a group of 7 boys and 6 girls, five persons are to be selected to form a team, so that at least 3 girls are there in the team. In how many ways can it be done?

 A) 427 B) 531 C) 651 D) 714

Explanation:

Given in the question that, there are 7 boys and 6 girls.

Team members = 5

Now, required number of ways in which a team of 5 having atleast 3 girls in the team =

2 140
Q:

The number of ways in which 8 distinct toys can be distributed among 5 children?

 A) 5P8 B) 5^8 C) 8P5 D) 8^5

Explanation:

As the toys are distinct and not identical,

For each of the 8 toys, we have three choices as to which child will receive the toy. Therefore, there are ${\mathbf{5}}^{\mathbf{8}}$ ways to distribute the toys.

Hence, it is ${\mathbf{5}}^{\mathbf{8}}$ and not ${\mathbf{8}}^{\mathbf{5}}$.

1 652
Q:

In how many different ways can the letters of the word 'THERAPY' be arranged so that the vowels never come together?

 A) 1440 B) 720 C) 2250 D) 3600

Explanation:

Given word is THERAPY.

Number of letters in the given word = 7

These 7 letters can be arranged in 7! ways.

Number of vowels in the given word = 2 (E, A)

The number of ways of arrangement in which vowels come together is 6! x 2! ways

Hence, the required number of ways can the letters of the word 'THERAPY' be arranged so that the vowels never come together = 7! - (6! x 2!) ways = 5040 - 1440 = 3600 ways.

1 377
Q:

In how many different ways can the letters of the word 'HAPPYHOLI' be arranged?

 A) 89,972 B) 90,720 C) 72,000 D) 81,000

Explanation:

The given word HAPPYHOLI has 9 letters

These 9 letters can e arranged in 9! ways.

But here in the given word letters H & P are repeated twice each

Therefore, Number of ways these 9 letters can be arranged is

5 531
Q:

How many words can be formed with or without meaning by using three letters out of k, l, m, n, o without repetition of alphabets.

 A) 60 B) 120 C) 240 D) 30

Explanation:

Given letters are k, l, m, n, o = 5

number of letters to be in the words = 3

Total number of words that can be formed from these 5 letters taken 3 at a time without repetation of letters =

7 520
Q:

The letters of the word PROMISE are to be arranged so that three vowels should not come together. Find the number of ways of arrangements?

 A) 4320 B) 4694 C) 4957 D) 4871

Explanation:

Given Word is PROMISE.

Number of letters in the word PROMISE = 7

Number of ways 7 letters can be arranged = 7! ways

Number of Vowels in word PROMISE = 3 (O, I, E)

Number of ways the vowels can be arranged that 3 Vowels come together = 5! x 3! ways

Now, the number of ways of arrangements so that three vowels should not come together

= 7! - (5! x 3!) ways = 5040 - 720 = 4320.

7 979
Q:

In how many different ways can the letters of the word 'POVERTY' be arranged ?

 A) 2520 B) 5040 C) 1260 D) None

Explanation:

The 7 letters word 'POVERTY' be arranged in $\mathbf{7}{\mathbit{P}}_{\mathbf{7}}$ ways = 7! = 5040 ways.

12 581
Q:

A decision committee of 5 members is to be formed out of 4 Actors, 3 Directors and 2 Producers. In how many ways a committee of 2 Actors, 2 Directors and 1 Producer can be formed ?

 A) 18 B) 24 C) 36 D) 32