# Problem of the Day: 1/21/13

Sara leaves Mathville at 2 PM, driving at 60 mph. Thomas leaves at 3 PM, driving at 80 mph. At what time will they meet?

Solution to yesterday’s problem:

Using the blanks method, we can represent the places like this:

____ ____ ____

Since there are seven runners vying for first, we’ll fill in the first blank:

__7__ ____ ____

One runner has already received first place, so there are six more runners for second, and likewise for third.

__7__ __6__ __5__

Multiplying these numbers, we get 210 ways.

Each way to order the runners is called a permutation. There is a formula to calculate the number of permutations given 2 things: the number of objects to select from, and the number of objects to actually choose. For instance, in this problem we were selecting 3 runners out of 7. We would write this as $_7 P_3$. The formula to use for permutations is

$_n P_r = \frac{n!}{(n-r)!}$

This formula bears resemblance to the blanks method. Once the numbers have been plugged in and the fraction simplified, you will also get 7 x 6 x 5.