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