How many different seven-digit numbers can be produced by rearranging the digits of ?

Source: mathcontest.olemiss.edu 10/1/2012

**SOLUTION**

Let’s pick a random digit other than zero among to be the leading digit. For example, let’s pick . Even though the digits look the same stand-alone, a in the unit place is different from a in the ten place which is different from a in the hundred place. For example, dollars is different from dollars which is different from dollars.

For this reason we will treat the and digits as different items when rearranging them to produce the seven-digit numbers. We categorize the search for the different numbers by the leading digits , and .

CASE 1: How many different seven-digit numbers can we form by having as the leading digit?

STEP 1. Find two places to put the remaining two digits

STEP 2. Find two places to put the two digits

STEP 3. Find one place to put the digit

STEP 4. Find one place to put the digit

Count of different seven-digit numbers having as the leading digit

CASE 2: How many different seven-digit numbers can we form by having as the leading digit?

STEP 1. Find three places to put the three 2 digits

STEP 2. Find one place to put the digit

STEP 3. Find two places to put the two 5 digits

Count of seven-digit numbers having 4 as the leading digit

CASE 3: How many different seven-digit numbers can we form by having as the leading digit?

STEP 1. Find one place to put the remaining digit

STEP 2. Find three places to put the three 2 digits

STEP 3. Find one place to put the digit

STEP 4. Find one place to put the digit

Count of different seven-digit numbers having as the leading digit

How many different seven-digit numbers can be produced by rearranging the digits of ?

**Answer**: .