## 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: $300$