In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?
The word MISSISSIPPI contains 11 letters in total in which S appears 4 times, I appear 4 times, P appears 2 times and M appears only once. So, the number of permutation of the word MISSISSIPPI is given as,
11 ! 4 ! 4 ! 2 !
Cancel the common factors from numerator and denominator by factorizing the greater term in form of factorials.
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 permutation is written as,
11 ! 4 ! 4 ! 2 ! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 ! 4 ! × 4 × 3 × 2 × 1 × 2 × 1 = 11 × 10 × 9 × 8 × 7 × 6 × 5 4 × 3 × 2 × 1 × 2 × 1 = 34650
If four I’s will occur together, then they will be considered as one. So, the total letters in the word becomes 8 which contains 4 S and 2 P. Then the number of permutation of the word becomes,
8 ! 4 ! 2 ! = 8 × 7 × 6 × 5 × 4 ! 4 ! × 2 × 1 = 840
Since total number ways to write the word are 34650 and the total number of ways that four I’s occurs together are 840.
The total number of ways that four I’s do not occur together can be calculated by subtracting the total number of ways four I’s occur together from total number of ways the word can be formed,
34650 − 840 = 33810
Thus, required number of permutations in which four I’s do not come together is 33810.
The word MISSISSIPPI contains 11 letters in total in which S appears 4 times, I appear 4 times, P appears 2 times and M appears only once. So, the number of permutation of the word MISSISSIPPI is given as,
Cancel the common factors from numerator and denominator by factorizing the greater term in form of factorials.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The permutation is written as,
If four I’s will occur together, then they will be considered as one. So, the total letters in the word becomes 8 which contains 4 S and 2 P. Then the number of permutation of the word becomes,
Since total number ways to write the word are 34650 and the total number of ways that four I’s occurs together are 840.
The total number of ways that four I’s do not occur together can be calculated by subtracting the total number of ways four I’s occur together from total number of ways the word can be formed,
Thus, required number of permutations in which four I’s do not come together is 33810.
