# Bar of the Week – Solution 4 –  Fall 2007

Pick 5 Combos

There are 792 possible combinations so the “over 790 statement” is correct.  Arby’s did a good job mathematically with this ad!

To count these without actually listing them, think of having eight boxes labeled with the 8 possible choices (Arby’s Melt, Medium Soft Drink, etc.) and five unmarked balls.  Each way of distributing the balls in the boxes gives exactly one of the possible combinations.  For example, putting two balls in the Arby’s Melt box, one in the Turnover box, one in the Small Shake box, and one in the Curly Fries box would correspond to the choice shown as “Matt’s Picks” in the advertisement.

So how many ways can we put 5 balls in 8 boxes?  Seven vertical dashes divide a blank line into 8 pieces corresponding to the eight boxes.  (We need to assume that the boxes are arranged in some definite order.  The order given in the advertisement would work just fine so that the first box is for Arby’s Melt, the second is for Medium Soft Drink, etc.)  Distributing the five balls in the eight boxes then corresponds to arranging 5 balls and 7 dashes in some definite order.  For example, Matt’s Picks would be:

o o |  | o |  |  |  o  |  o  |

which represents two balls in the first box (Arby’s Melt), one in the third (Turnover), one in the sixth (Small Shake), and one in the seventh (Curly Fries).

The number of ways of arranging 5 balls and 7 dashes is “12 choose 5” (we’re choosing the locations of the 5 balls from the 12 possible spots) and that is 12x11x10x9x8/(1x2x3x4x5)  which is 792.