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Permutations And Combinations

Question
CBSEENMA11014164

In how many ways can the letters of word ‘ASSASSINATION’ be arranged so that all the S’s are together?

Solution

Number of letters:
                      A → 3,  S → 4
                      I → 2,   N → 2
                      T → 1,   O → 1
Total letters 13.
Tie the 4 S's.
Number of permutations = fraction numerator straight P presuperscript 4 subscript 4 over denominator 4 factorial end fraction space equals space fraction numerator 4 factorial over denominator 4 factorial end fraction equals 1
Mix with remaining to give:
[3 (A) + 2 (I) + 2 (N) + 1 (T) + 1 (O)] + 1 = 10.
Number of arrangements = fraction numerator 10 factorial over denominator 3 factorial space 2 factorial space 2 factorial end fraction space equals space fraction numerator 10 cross times 9 cross times 8 cross times 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 over denominator 3 cross times 2 cross times 1 cross times 2 cross times 1 cross times 2 cross times 1 end fraction equals 151200.
Total number of arrangements = 1 x 151200 = 151200.

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