4064465

You are given the following seven-digit number: 4064465. How many different seven-digit numbers can be made by rearranging the digits of 4064465?
Source: mathcontest.olemiss.edu 4/30/2012

SOLUTION
Consider the set of seven digits \left \{0,4,4,4,5,6,6\right \}. We have seven places to put the seven digits. We can put the digits \left \{4,4,4,5,6,6\right \} in any place but we cannot put digit \left \{0\right \} in the leading place because we want to make seven-digit numbers.

Though we can begin with any digit, let’s start with digit \left \{0\right \} 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 \left \{0\right \}?
\binom{6}{1}

There are six places remaining to put the remaining six digits \left \{4,4,4,5,6,6\right \}.
How many ways can we choose three places out of six to put the digits \left \{4,4,4\right \}?
\binom{6}{3}

There are three places remaining to put the remaining three digits \left \{5,6,6\right \}.
How many ways can we choose one place out of three to put the digit \left \{5\right \}?
\binom{3}{1}

There are two places remaining to put the remaining two digits \left \{6,6\right \}.
How many ways can we choose two places out of two to put the digits \left \{6,6\right \}?
\binom{2}{2}

Total ways = \binom{6}{1}\times\binom{6}{3}\times\binom{3}{1}\times\binom{2}{2}
=6\times 20\times 3\times 1
=360

Answer: 360

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About mvtrinh

Retired high school math teacher.
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