NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

In how many different ways can the letters of the word "CORPORATION" be arranged so that all the vowels always come together?

Choose the correct answer:

Explanation

Solution

1. Analyze the word "CORPORATION":

Total letters = $11$
Vowels: $O, O, A, I, O$ (Total $5$ vowels)
Consonants: $C, R, P, R, T, N$ (Total $6$ consonants)

2. Group the vowels:

Since all vowels must come together, we treat $(OOAIO)$ as one single unit.

New count of entities = $6$ consonants + $1$ vowel unit = $7$ entities.

3. Arrange the 7 entities:

In these 7 entities, the letter ' $R$ ' is repeated twice.

\text{Ways to arrange 7 entities} = \frac{7!}{2!}

\frac{5040}{2} = 2520

4. Arrange the vowels within their unit:

In the vowel unit $(OOAIO)$ , there are $5$ letters where ' $O$ ' is repeated three times.

\text{Ways to arrange vowels} = \frac{5!}{3!}

\frac{120}{6} = 20

5. Total number of ways:

\text{Total ways} = 2520 \times 20

\text{Total ways} = 50400