The letters of the word SOCIETY are arranged | vedclass

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(D) The word $SOCIETY$ contains $7$ distinct letters: $S, O, C, I, E, T, Y$.The total number of ways to arrange these $7$ letters is $n(S) = 7!$.The v

Vedclass · Accepted Answer

$\frac{1}{7}$

Vedclass · Answer

(D) The word $SOCIETY$ contains $7$ distinct letters: $S, O, C, I, E, T, Y$.The total number of ways to arrange these $7$ letters is $n(S) = 7!$.The vowels in the word are $O, I, E$ (total $3$ vowels).To find the number of arrangements where the $3$ vowels come together,we treat the group $(O, I, E)$ as a single unit.Now,we have $5$ units to arrange: $(OIE), S, C, T, Y$.These $5$ units can be arranged in $5!$ ways.Within the group $(OIE)$,the $3$ vowels can be arranged among themselves in $3!$ ways.Thus,the number of favorable outcomes is $n(A) = 5! 	imes 3!$.The probability $P(A) = \frac{n(A)}{n(S)} = \frac{5! 	imes 3!}{7!} = \frac{5! 	imes 6}{7 	imes 6 	imes 5!} = \frac{6}{42} = \frac{1}{7}$.

The letters of the word $SOCIETY$ are arranged in a row. What is the probability that the $3$ vowels come together?

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