# University of South Carolina High School Math Contest/1993 Exam/Problem 14

## Problem

How many permutations of 1, 2, 3, 4, 5, 6, 7, 8, 9 have:

- 1 appearing somewhere to the left of 2,
- 3 somewhere to the left of 4, and
- 5 somewhere to the left of 6?

For example, 8 1 5 7 2 3 9 4 6 would be such a permutation.

## Solution

There are (the factorial of 9) total permutations of the elements of that set. 1 is to the left of 2 in exactly half of these. 3 is also to the left of 4 in exactly half of the permutations, and 5 is to the left of 6 in exactly half of the permutations. These three events are totally independent of each other, so the number we want to calculate is which is answer choice .

Alternatively, note that we can choose places for the 1 and 2, then places for the 3 and 4, then places for the 5 and 6, and the arrange the 7, 8 and 9 in ways, giving us a total of .