I was working through this year’s MathCounts State Level exam, and found 2 problems involving factorials. Your reflex when seeing a factorial problem is to see how you can factor the expression. For example we are asked to find the value of this expression:

Notice that the 2 terms in the numerator are both products of and the 2 terms in the denominator are both products of . So we can factor these both out:

If you notice that and both have a factor of then the fraction is much easier to evaluate.

Another problem on the same exam asks us to solve for :

Again you can factor from the 2 terms on the left hand side:

To finish out this problem we can find the prime factorization of and match that up to the product of consecutive integers.

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## Published by mathproblemsolvingskills

I coach students preparing for MathCounts and the AMCs, and I teach using curricula published by Art of Problem Solving.
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