How many different ways can letters be arranged so that vowels always come together?

Answer [Detailed Solution Below]

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Given:

The word is DRASTIC.

Calculation:

The vowels in the word = A, I.

The number of vowel is 2.

The number of consonants = 5 [D R S T C]

To find the total number of ways we have to assume two vowel as one letter like [A I] D R S T C

The number of ways to arrange these 6 letters = 6! 

And total number of ways to arrange vowels = 2!

So the total number of ways = 6! × 2! = 1440

∴The number of ways that the vowels always come together is 1440. 

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2] In what ways the letters of the word "PUZZLE" can be arranged to form the different new words so that the vowels always come together?

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 =

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

3] In what ways can a group of 6 boys and 2 girls be made out of the total of 7 boys and 3 girls?

Answer: C

Answer with the explanation:

We know that nCr = 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] =

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

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

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

nCr=

So, [7C3 * 6C2] + [7C4 * 6C1] + [7C5]
Or,

Or, 525 +210+21 = 756

So, there are 756 ways to form a committee.

5] A box contains 2 red balls, 3 black balls, and 4 white balls. Find the number of ways by which 3 balls can be drawn from the box in which at least 1 black ball should be present.

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].

In the below solved problem, every thing is okay, but if we have $4$ consonants then why we are giving $5!$? and is this a combination problem? how to distinguish?

Question: In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?

Answer: The word 'OPTICAL' contains $7$ different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL [OIA]. Now, $5$ letters can be arranged in $5! = 120$ ways. The vowels [OIA] can be arranged among themselves in $3! = 6$ ways. Required number of ways $= [120*6] = 720$.

Exercise :: Permutation and Combination - General Questions

11. 

In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

Answer: Option A

Explanation:

Required number of ways = [7C5 x 3C2] = [7C2 x 3C1] =

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