You are given the following seven-digit number: . How many different seven-digit numbers can be made by rearranging the digits of ?

Source: mathcontest.olemiss.edu 4/30/2012

**SOLUTION**

Consider the set of seven digits . We have seven places to put the seven digits. We can put the digits in any place but we cannot put digit in the leading place because we want to make seven-digit numbers.

Though we can begin with any digit, let’s start with digit which occupies one place.

How many ways can we choose one place out of six (not seven because no leading zero) to put the digit ?

There are six places remaining to put the remaining six digits .

How many ways can we choose three places out of six to put the digits ?

There are three places remaining to put the remaining three digits .

How many ways can we choose one place out of three to put the digit ?

There are two places remaining to put the remaining two digits .

How many ways can we choose two places out of two to put the digits ?

Total ways =

**Answer**: