The Perfect Pack

I have a quirk.

Okay, truth be known, I have many. But this post is only going to elaborate on a particular mathematical quirk that I have, which involves eating M&M’s. (Most of my other quirks aren’t interesting enough to warrant a blog post. And those that are probably shouldn’t be publicized.)

I have to eat M&M’s in pairs of the same color. I place two in my mouth at a time, and I chew one on each side. I can’t eat them one at a time, and I can’t eat two M&M’s of different colors at the same time. It’s a balance thing. I’ve done this since I was a kid, and whether it’s just a bad habit or a deeply engrained compulsion, I don’t worry about it too much. Sure, it’s a little weird, but on the OCD continuum, it’s not a big deal. I mean, it’s not like I use a ruler to ensure that stamps are placed exactly the same distance from both sides of the envelope. (Though I have considered it.)

So, the math of this. I have been searching for the “perfect pack of M&M’s,” one in which there is an even number of every color. That way, I won’t have one leftover after I eat the others in pairs. I don’t know how many packs of M&M’s I’ve eaten in my life, but I’ve yet to find a perfect pack. Consequently, it would seem that the experimental probability of such an occurrence is 0. But what is the theoretical probability?

Stated formally, here’s the problem:

A standard pack of M&M’s contains pieces of six different colors. What is the probability that there will be an even number of M&M’s of each of the six colors?

I’ll post my solution in a couple days. In the meantime, here are two math jokes about M&M’s:

How many mathematicians does is take to make a batch of chocolate chip cookies?
Two. One to mix the batter, and one to peel the M&M’s.

How do you keep a math graduate student occupied for hours?
Ask him to alphabetize a bag of M&M’s.