# Problem of the Day: 2/22/13

A display table at a store has 20 ties. A equal number contain red and blue, but 8 ties are striped red and blue. What is the probability of picking a tie with only red?

Solution to yesterday’s problem:

This is one of those problems that you just have to use your logic to solve. Math competitions like AMC and Mathcounts love these problems because they would require titanic effort to compute manually (calculating 105 factorial). In this case, you’ll need to find the number of spots in the expansion of the factorial (105 x 104 x …) where a zero is introduced.

Which digit would likely introduce a zero? Of course, any multiple of ten will add a zero to the end of the product. However, we also know that multiples of five can add a zero if the other number is even. Luckily, we know that the number will be even because we definitely multiply by an even number somewhere. So every multiple of five adds a zero – that’s 105/5 = 21 zeros.

However, we’re not finished yet! We need to account for the factor of 100. We know multiplying something by 100 adds 2 zeros to the product, and we only accounted for 1 of the zeros. So let’s just tack a zero on the end of our number, and we have a final answer of 22 zeros.