Answer: D

Answer with the explanation:

The word RUMOUR consists of 6 words in which R and U are repeated twice.
Therefore, the required number of permutations =
Or, = 180

Hence, 180 words can be formed by arranging the word RUMOUR.

Answer: D

Answer with the explanation:

The word PUZZLE has 6 different letters.

As per the question, the vowels should always come together.
Now, let the vowels UE as a single entity.
Therefore, the number of letters is 5 (PZZL = 4 + UE = 1)
Since the total number of letters = 4+1 = 5
So the arrangement would be in ⁵P₅ = = = 5! = 5*4*3*2*1 = 120 ways.

Note: we know that 0! = 1

Now, the vowels UE can be arranged in 2 different ways, i.e., ²P₂ = 2! = 2*1 = 2 ways

Hence, the new words, which can be formed after rearranging the letters = 120 *2 = 240

As we known z is occurring twice in the word ‘PUZZLE’ so we will divide the 240 by 2.

So, the no. of permutation will be = 240/2 = 120

Answer: C

Answer with the explanation:

We know that ⁿC_r = ⁿC_(n-r)

The combination of 6 boys out of 7 and 2 girls out of 3 can be represented as ⁷C₆ + ³C₂
Therefore, the required number of ways = ⁷C₆ * ³C₂ = ⁷C(7-6) * ³C_(3-2) = = 21

Hence, in 21 ways the group of 6 boys and 2 girls can be made.

Answer: C

Answer with the explanation:

We may have 5 men only, 4 men and 1 woman, and 3 men and 2 women in the committee.

So, the combination will be

as we know that

ⁿC_r=

So, (⁷C₃ * ⁶C₂) + (⁷C₄ * ⁶C₁) + (⁷C₅)
Or, + +

Or, 525 +210+21 = 756

So, there are 756 ways to form a committee.

Answer: A

Answer with the explanation:

The possible combination could be (1 black ball and 2 non-black balls), (2 black balls and 1 non- black ball), and (only 3 black balls).

Therefore the required number of combinations = (³C₁ * ⁶C₂) + (³C₂ * ⁶C₁) + (³C₃)
r, + + = 45+18+1 = 64