EXAMPLE 44 Permutations with nondistinct items

How many distinct strings of letters can we make by using all the letters in the word STATISTICS?

Solution

Each string will be 10 letters long and include 3 S's, 3 T's, 2 I's, 1 A, and 1 C. The 10 positions shown here need to be filled.

The string-forming process is as follows:

  • Step 1 Choose the positions for the three S's.
  • Step 2 Choose the positions for the three T's.
  • Step 3 Choose the positions for the two I's.
  • Step 4 Choose the position for the one A.
  • Step 5 Choose the position for the one C.

There are ways to place the three S's in Step 1. Once Step 1 is done, seven slots are left, leaving positions for the three T's. Once Step 2 is done, four slots are left, so there are ways to place the two I's. Once Step 3 is done, only two slots are left, so there are only ways to position the A. Finally, there is only way to place the C.

Putting Steps 1–5 together, we calculate the number of distinct letter strings as

There are 50,400 distinct strings of letters that can be made using the letters in the word STATISTICS.