Using all the digits and , how many distinct integers greater than can be formed?

Source: NCTM Mathematics Teacher, January 2006

**Solution**

The first digit must be a or a .

Case 1: _ _ _ _ _ _

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 in a four-step process

1) choose a subset of positions for the two ’s

2) choose a subset of positions for the two ’s

3) choose a subset of position for the

4) choose a subset of position for the

Case 2: _ _ _ _ _ _

Similarly, we construct an ordering for the digits

1) choose a subset of position for the

2) choose a subset of positions for the ’s

3) choose a subset of position for the

4) choose a subset of position for the

distinct integers greater than can be formed using all the digits , and .

**Answer**: