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.

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , . Bookmark the permalink.

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