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.