Suppose that you placed the letters in Example 9.5.11 into positions in the following order: first the M. then the I’s, then the S’s, and then the P’s. Show that you would obtain the same answer for the number of distinguishable orderings.

To prove:

The answer is same for the number of distinguishable orderings for the word MISSISSIPPI.

Given information:

Consider various ways of ordering the letters in the word MISSISSIPPI :

IIMSSPISSIP, ISSSPMIIPIS, PIMISSSSIIP, and so on.

Suppose that you placed the letters in MISSISSIPPIinto positions in the following order: first the M, then the I’s, then the S’s, and then the P’s.

Proof:

Since the order in which we select the positions for the same letter doesn’t matter, we should use the definition of a combination.

The word MISSISSIPPI contains 11 letters (positions) of which 1 is an M, 2 are P’s, 4 are I’s and 4 are S’s.

First, we select 1 of the 11 positions for the M, next we select 2 of the remaining 11−1=10 positions for the P’s, next we select 4 of the remaining 10−2=8 positions for the I’s and finally we select 4 of the remaining 8−4=4 positions for the S’s.

Letter M: ( 11 1 )=11!1!( 11−1)!=11!1!10!=11​ waysLetter P: ( 10 2 )=10!2!