In how many ways can 5 letters be selected fr | vedclass

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Question

(A) The word $INDEPENDENT$ contains $11$ letters: $3N, 3E, 2D, 1I, 1P, 1T$. The distinct letters are ${N, E, D, I, P, T}$.We need to select $5$ letter

Vedclass · Accepted Answer

$72$

Vedclass · Answer

(A) The word $INDEPENDENT$ contains $11$ letters: $3N, 3E, 2D, 1I, 1P, 1T$. The distinct letters are ${N, E, D, I, P, T}$.We need to select $5$ letters. The possible cases are:$(i)$ All $5$ letters are distinct: We choose $5$ from ${N, E, D, I, P, T}$. Number of ways = $^6C_5 = 6$.$(ii)$ $2$ are alike (of one kind) and $3$ are distinct: We choose $1$ pair from ${N, E, D}$ and $3$ distinct letters from the remaining $5$ types. Number of ways = $^3C_1 	imes ^5C_3 = 3 	imes 10 = 30$.$(iii)$ $3$ are alike (of one kind) and $2$ are distinct: We choose $1$ triplet from ${N, E}$ and $2$ distinct letters from the remaining $5$ types. Number of ways = $^2C_1 	imes ^5C_2 = 2 	imes 10 = 20$.$(iv)$ $3$ are alike (of one kind) and $2$ are alike (of another kind): We choose $1$ triplet from ${N, E}$ and $1$ pair from the remaining $2$ types. Number of ways = $^2C_1 	imes ^2C_1 = 2 	imes 2 = 4$.$(v)$ $2$ are alike (of one kind),$2$ are alike (of another kind),and $1$ is distinct: We choose $2$ pairs from ${N, E, D}$ and $1$ distinct letter from the remaining $4$ types. Number of ways = $^3C_2 	imes ^4C_1 = 3 	imes 4 = 12$.Total number of ways = $6 + 30 + 20 + 4 + 12 = 72$.

In how many ways can $5$ letters be selected from the letters of the word $INDEPENDENT$?

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