In how many ways can the letters of the word | vedclass

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(A) The word $PERMUTATIONS$ contains $12$ letters: $P, E, R, M, U, T, A, T, I, O, N, S$. The letter $T$ repeats $2$ times.There are $12$ positions. If

Vedclass · Accepted Answer

$25401600$

Vedclass · Answer

(A) The word $PERMUTATIONS$ contains $12$ letters: $P, E, R, M, U, T, A, T, I, O, N, S$. The letter $T$ repeats $2$ times.There are $12$ positions. If we place $P$ at position $i$,then $S$ must be at position $i+5$ to have $4$ letters between them.The possible pairs of positions $(i, i+5)$ are $(1, 6), (2, 7), (3, 8), (4, 9), (5, 10), (6, 11), (7, 12)$. There are $7$ such pairs.Since $P$ and $S$ can be interchanged,there are $7 	imes 2 = 14$ ways to place $P$ and $S$.The remaining $10$ letters can be arranged in $\frac{10!}{2!}$ ways because $T$ repeats twice.Total arrangements $= 14 	imes \frac{10!}{2!} = 7 	imes 10! = 7 	imes 3628800 = 25401600$.

In how many ways can the letters of the word $PERMUTATIONS$ be arranged if there are always $4$ letters between $P$ and $S$?

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