Question

If the number of arrangements of the letters of the word 'MISSISSIPPI' if all the S's and P's are separated is

Answer 1

3375

Answer 2

The problem asks for the number of arrangements of the letters of the word 'MISSISSIPPI' such that all the S's are separated and all the P's are separated.

The letters in 'MISSISSIPPI' and their frequencies are: M: 1 I: 4 S: 4 P: 2

Total letters = 1 + 4 + 4 + 2 = 11.

The conditions are:

1. No two S's are adjacent (SS is not allowed).
2. No two P's are adjacent (PP is not allowed).

We use the "gap method" to solve this problem.

Step 1: Arrange the letters that are neither S nor P.

These letters are M and I (four times). The number of arrangements of M, I, I, I, I is: $N_1 = \frac{5!}{1! \times 4!} = \frac{120}{1 \times 24} = 5$.

Let's consider one such arrangement, for example, M I I I I. These 5 letters create 6 possible gaps where the S's and P's can be placed to ensure separation. _ M _ I _ I _ I _ I _ Let these gaps be $G_1, G_2, G_3, G_4, G_5, G_6$.

Step 2: Place the 4 S's such that no two S's are adjacent.

To ensure no two S's are adjacent, we must place each S in a different gap. We have 6 gaps and need to choose 4 of them for the 4 S's. The number of ways to choose 4 gaps out of 6 is $\binom{6}{4}$. $\binom{6}{4} = \frac{6!}{4!2!} = \frac{6 \times 5}{2 \times 1} = 15$. Once the gaps are chosen, the S's are identical, so there's only 1 way to place them.

After placing the S's, we have a configuration like: S M S I S I S I I (if S's are in gaps 1, 2, 3, 4) Now, we need to identify the new gaps created by these letters (M, I, and S) to place the P's. The total number of letters arranged so far is 5 (M, I's) + 4 (S's) = 9 letters. These 9 letters create 10 gaps. For example, if we arranged S's and M, I's as S M S I S I S I I. The gaps are: _ S _ M _ S _ I _ S _ I _ S _ I _ I _ There are (9+1) = 10 gaps.

Step 3: Place the 2 P's such that no two P's are adjacent.

To ensure no two P's are adjacent, we must place each P in a different gap. We have 10 gaps and need to choose 2 of them for the 2 P's. The number of ways to choose 2 gaps out of 10 is $\binom{10}{2}$. $\binom{10}{2} = \frac{10!}{2!8!} = \frac{10 \times 9}{2 \times 1} = 45$. Once the gaps are chosen, the P's are identical, so there's only 1 way to place them.

Step 4: Calculate the total number of arrangements.

The total number of arrangements is the product of the number of ways at each step: Total arrangements = (Arrangements of M, I's) $\times$ (Ways to place S's) $\times$ (Ways to place P's) Total arrangements = $N_1 \times \binom{6}{4} \times \binom{10}{2}$ Total arrangements = $5 \times 15 \times 45$ Total arrangements = $75 \times 45$ Total arrangements = $3375$.