How many different four-letter words can be f | vedclass

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Question

(A) The word $MEDITERRANEAN$ consists of $13$ letters: $M, E, D, I, T, E, R, R, A, N, E, A, N$. The frequency of letters is: $E:3, R:2, A:2, N:2, M:1,

Vedclass · Accepted Answer

$59$

Vedclass · Answer

(A) The word $MEDITERRANEAN$ consists of $13$ letters: $M, E, D, I, T, E, R, R, A, N, E, A, N$. The frequency of letters is: $E:3, R:2, A:2, N:2, M:1, D:1, I:1, T:1$. Total distinct letters available are ${M, E, D, I, T, R, A, N}$,which is $8$ distinct types.
We need to form a $4$-letter word where the first letter is $E$ and the last is $R$. The word structure is $E . . R$.
We need to fill the $2$ middle positions using the remaining letters. The available letters after fixing $E$ and $R$ are: $E:2, R:1, A:2, N:2, M:1, D:1, I:1, T:1$. Total $11$ letters remain.
Case $1$: The two middle letters are the same. The letters that appear at least twice are $E, A, N$. We can choose one of these $3$ pairs. Number of ways $= 3$.
Case $2$: The two middle letters are different. We need to choose $2$ distinct letters from the $8$ available types ${M, E, D, I, T, R, A, N}$. The number of ways to choose and arrange $2$ distinct letters is $P(8, 2) = 8 	imes 7 = 56$.
Total number of words $= 56 + 3 = 59$.

How many different four-letter words can be formed (the words need not be meaningful) using the letters of the word $MEDITERRANEAN$ such that the first letter is $E$ and the last letter is $R$?

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