Numbers Greater Than 5,000,000

Using all the digits 3,4,5,5,5,6 and 6, how many distinct integers greater than 5,\!000,\!000 can be formed?
Source: NCTM Mathematics Teacher, January 2006

Solution
The first digit must be a 5 or a 6.
Case 1: 5 _ _ _ _ _ _
Since copies of the same digits cannot be distinguished from one another, once we have positions to put them in we can copy them into the positions in any order. We construct an ordering for the digits 3,4,5,5,6,6 in a four-step process
1) choose a subset of 2 positions for the two 6’s
2) choose a subset of 2 positions for the two 5’s
3) choose a subset of 1 position for the 4
4) choose a subset of 1 position for the 3
\dbinom{6}{2}\dbinom{4}{2}\dbinom{2}{1}\dbinom{1}{1}=180
Case 2: 6 _ _ _ _ _ _
Similarly, we construct an ordering for the digits 3,4,5,5,5,6
1) choose a subset of 1 position for the 6
2) choose a subset of 3 positions for the 5’s
3) choose a subset of 1 position for the 4
4) choose a subset of 1 position for the 3
\dbinom{6}{1}\dbinom{5}{3}\dbinom{2}{1}\dbinom{1}{1}=120
180+120=300
300 distinct integers greater than 5,\!000,\!000 can be formed using all the digits 3,4,5,5,5,6, and 6.

Answer: 300

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About mvtrinh

Retired high school math teacher.
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