The number of ways in which all the letters o | vedclass

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(C) The word $COMBINATION$ consists of $11$ letters: $C, O, M, B, I, N, A, T, I, O, N$. The vowels are $O, I, A, I, O$ (total $5$ vowels). The consona

Vedclass · Accepted Answer

$75600$

Vedclass · Answer

(C) The word $COMBINATION$ consists of $11$ letters: $C, O, M, B, I, N, A, T, I, O, N$. The vowels are $O, I, A, I, O$ (total $5$ vowels). The consonants are $C, M, B, N, N$ (total $6$ consonants). Since the vowels must come together,we treat the group $(O, I, A, I, O)$ as a single entity. Now,we have $6$ consonants + $1$ entity = $7$ items to arrange. The number of arrangements of these $7$ items,where $N$ repeats $2$ times,is $\frac{7!}{2!} = \frac{5040}{2} = 2520$. Within the vowel group $(O, I, A, I, O)$,there are $5$ letters where $O$ repeats $2$ times and $I$ repeats $2$ times. The number of arrangements of these vowels is $\frac{5!}{2! 	imes 2!} = \frac{120}{4} = 30$. The total number of ways is $2520 	imes 30 = 75600$.

The number of ways in which all the letters of the word $COMBINATION$ can be arranged so that the vowels always come together is:

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