In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together?
The word ASSASSINATION contains 13 letters in total in which S appears 4 times, A appear 3 times, N appears 2 times and I appear 2 times.
When the four S’s will occur together, then they will be considered as one. Thus, the total letters in the word becomes 10 which contains 3 A, 2 N and 2 I. Then the number of permutation of the word becomes,
10 ! 3 ! 2 ! 2 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The term becomes,
10 ! 3 ! 2 ! 2 ! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 ! 3 ! 2 × 1 × 2 × 1 = 151200
Thus, the total number of ways that four S’s occur together is 151200.
The word ASSASSINATION contains 13 letters in total in which S appears 4 times, A appear 3 times, N appears 2 times and I appear 2 times.
When the four S’s will occur together, then they will be considered as one. Thus, the total letters in the word becomes 10 which contains 3 A, 2 N and 2 I. Then the number of permutation of the word becomes,
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The term becomes,
Thus, the total number of ways that four S’s occur together is 151200.
