## 2502254

How many different seven-digit numbers can be produced by rearranging the digits of $2502254$?
Source: mathcontest.olemiss.edu 10/1/2012

SOLUTION
Let’s pick a random digit other than zero among $0,2,2,2,4,5,5$ to be the leading digit. For example, let’s pick $2$. Even though the $2$ digits look the same  stand-alone, a $2$ in the unit place is different from a $2$ in the ten place which is different from a $2$ in the hundred place. For example, $2$ dollars is different from $22$ dollars which is different from $222$ dollars.
For this reason we will treat the $2$ and $5$ 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 $2, 4$, and $5$.
CASE 1: How many different seven-digit numbers can we form by having $2$ as the leading digit?
STEP 1. Find two places to put the remaining two $2$ digits
$\binom{6}{2}=15$
STEP 2. Find two places to put the two $5$ digits
$\binom{4}{2}=6$
STEP 3. Find one place to put the $4$ digit
$\binom{2}{1}=2$
STEP 4. Find one place to put the $0$ digit
$\binom{1}{1}=1$
Count of different seven-digit numbers having $2$ as the leading digit
$15\times 6\times 2\times 1=180$

CASE 2: How many different seven-digit numbers can we form by having $4$ as the leading digit?
STEP 1. Find three places to put the three 2 digits
$\binom{6}{3}=20$
STEP 2. Find one place to put the $0$ digit
$\binom{3}{1}=3$
STEP 3. Find two places to put the two 5 digits
$\binom{2}{2}=1$
Count of seven-digit numbers having 4 as the leading digit
$20\times 3\times 1=60$

CASE 3: How many different seven-digit numbers can we form by having $5$ as the leading digit?
STEP 1. Find one place to put the remaining $5$ digit
$\binom{6}{1}=6$
STEP 2. Find three places to put the three 2 digits
$\binom{5}{3}=10$
STEP 3. Find one place to put the $0$ digit
$\binom{2}{1}=2$
STEP 4. Find one place to put the $4$ digit
$\binom{1}{1}=1$
Count of different seven-digit numbers having $5$ as the leading digit
$6\times 10\times 2\times 1=120$

How many different seven-digit numbers can be produced by rearranging the digits of $2502254$?
$180+60+120=360$

Answer: $360$.