The Probability of All 'S's Coming Together in the Word 'MISSISSIPPI'

Understanding the probability of specific arrangements in words like 'MISSISSIPPI' is a classic problem in combinatorics and discrete mathematics. This article breaks down the process to find the probability that all the 'S's in the word 'MISSISSIPPI' are arranged together.

Step 1: Total Arrangements of the Letters

The word 'MISSISSIPPI' consists of 11 letters, where there are 4 'I's, 4 'S's, 2 'P's, and 1 'M'. The total number of arrangements of these letters can be calculated using the formula for permutations of multiset:

[ text{Total arrangements} frac{n!}{n_1! cdot n_2! cdot n_3! cdots} ]

Here, ( n ) is the total number of letters, and ( n_1, n_2, n_3, ldots ) are the frequencies of the distinct letters.

For 'MISSISSIPPI', the total number of arrangements is:

[ text{Total arrangements} frac{11!}{1! cdot 4! cdot 4! cdot 2!} ]

Calculating this:

[ 11! 39916800 ] [ 1! 1, 4! 24, 2! 2 ] [ 4! cdot 4! cdot 2! 24 cdot 24 cdot 2 1152 ] [ text{Total arrangements} frac{39916800}{1152} 34650 ]

Step 2: Arrangements with 'S's Together

If we consider all the 'S's as a single unit or block, we can represent:

[ text{Block of Ss SSSS, M, 4 I's, 2 P's} ]

This gives us a total of 8 units to arrange: S, M, I, I, I, I, P, P. The arrangements of these 8 units are given by:

[ text{Arrangements with Ss together} frac{8!}{1! cdot 4! cdot 2!} ]

Calculating this:

[ 8! 40320 ] [ 4! cdot 4! cdot 2! 24 cdot 24 cdot 2 1152 ] [ text{Arrangements with Ss together} frac{40320}{1152} 35 cdot 24 840 ]

Step 3: Probability that All 'S's Come Together

Now, the probability that all the 'S's come together is given by the ratio of the number of favorable arrangements to the total arrangements:

[ P_{text{all Ss together}} frac{text{Arrangements with Ss together}}{text{Total arrangements}} frac{840}{34650} ]

Calculating the probability:

[ P_{text{all Ss together}} frac{840}{34650} approx 0.0242 ]

Final Result:

Thus, the probability that all the 'S's in the word 'MISSISSIPPI' are all together is approximately 0.0242 or 2.42%. This probability is a direct application of fundamental principles in combinatorics and provides a clear illustration of how specific conditions can influence the outcome in complex permutations.