## Thinking Mathematically (6th Edition)

In how many ways can you choose the first letter? Is it 7 (seven letters in all)? Or is it 5 (5 distinct letters? Once the first letter is chosen, in how many ways can we choose the second? We have a problem here, because this selection depends whether we picked one of the duplicates for the first or some other letter. We can see that the Fundamental Counting Principle is not appropriate for this problem. Instead, we use Permutations of Duplicate Items (see page 699), The number of permutations of $n$ items, where $p$ items are identical, $q$ items are identical, $r$ items are identical, and so on, is $\displaystyle \frac{n!}{p!q!r!\ldots}$ Here, the total is $\displaystyle \frac{7!}{2!2!}=\frac{7\times 6\times 5\times(4)\times 3\times 2\times 1}{(2\times 2)}$ $=7\times 6\times 5\times 3\times 2=1260$