The number of ways of arranging all the lette | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The word "$COMBINATIONS$" has $12$ letters: $C, O, M, B, I, N, A, T, I, O, N, S$.Consonants: $C, M, B, N, N, T, S$ ($7$ letters,with $N$ repeating

Vedclass · Accepted Answer

$\frac{7!6!}{(2!)^4}$

Vedclass · Answer

(A) The word "$COMBINATIONS$" has $12$ letters: $C, O, M, B, I, N, A, T, I, O, N, S$.Consonants: $C, M, B, N, N, T, S$ ($7$ letters,with $N$ repeating twice).Vowels: $O, I, A, I, O$ ($5$ letters,with $O$ repeating twice and $I$ repeating twice).First,arrange the $7$ consonants in a circle. The number of ways to arrange $n$ items in a circle is $(n-1)!$. Since $N$ repeats twice,the number of ways is $\frac{(7-1)!}{2!} = \frac{6!}{2!}$.There are $7$ gaps created between these $7$ consonants. We need to place the $5$ vowels in these $7$ gaps such that no two vowels are together. The number of ways to choose $5$ gaps out of $7$ is ${ }^{7}C_{5}$.The $5$ vowels can be arranged in these $5$ chosen gaps in $5!$ ways. Since $O$ and $I$ each repeat twice,the number of arrangements is $\frac{5!}{2!2!}$.Total number of ways $= \frac{6!}{2!} 	imes { }^{7}C_{5} 	imes \frac{5!}{2!2!} = \frac{6!}{2!} 	imes \frac{7!}{5!2!} 	imes \frac{5!}{2!2!} = \frac{7!6!}{(2!)^4}$.

The number of ways of arranging all the letters of the word "$COMBINATIONS$" around a circle so that no two vowels come together is

Similar Questions

The number of ways in which $3$ boys and $4$ girls can be seated around a circular table if no two boys sit together is

In how many ways can $4$ beads be selected and arranged in a necklace from $6$ distinct beads?

The number of ways in which $5$ boys and $4$ girls can sit around a circular table such that no two girls sit together is:

In how many ways can $5$ keys be arranged in a ring?

Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a cap of one of the three colours red,blue and green,then the number of ways of distributing the caps such that the persons seated in adjacent seats get different coloured caps,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module