Question

How many arrangements of length n where 1 ≤ n ≤ 8 can be formed from the letters A, A, B, C, C, C, D, E where

(a) both A’s are adjacent?

(b) the string starts or ends with A?

(c) you use (exactly) 4 letters from the list?

Answer 1

A, A, B, C, C, C, D, E

(a) both A’s are adjacent?

Consider, 7 letters ( AA being one)

The number of arrangements = 7! / ( 3! ) ( C is repeated thrice )

The number of arrangements = 7! / ( 3! )

The number of arrangements = 840

(b) the string starts or ends with A?

The number of arrangements where string start with A = 2*7!/(3!) ( 2 A's are there and remaining to be arranged in 7! ways , C repeated thrice )

The number of arrangements where string start with A = 840*2 = 1680

The number of arrangements where the string ends with A = 2*7!/(3!) ( 2 A's are there and remaining to be arranged in 7! ways , C repeated thrice )

The number of arrangements where the string ends with A = 840*2 = 1680

The number of arrangements where strings or ends with A = 1680 + 1680 = 3360

(c) you use (exactly) 4 letters from the list?

Selecting 4 unique letters out of 5 = 5C4*(3C1)*(2C1) ( 3 C's, 2 A's )

Selecting 4 unique letters out of 5 = 5*3*2 = 30