Bunuel wrote:

Mel and Nora share a total of three red marbles, two green marbles, and one blue marble. In how many ways can Mel and Nora divide the marbles between themselves, if it is not necessary for each of them to get at least one marble?

A. \(6\)

B. \(18\)

C. \(24\)

D. \(36\)

E. \(72\)

Official solution from

Veritas Prep.

Combinatorics questions, like this one, so often come down to a matter of approach. That is, we can tell the story of this problem in different ways, each of which leads to a valid solution, but some of which lead to far easier solutions than others.

For instance, we might view this problem as “Mel could get

zero,

one,

two,

three,

four,

five, or

six marbles, in a variety of color combinations” and then calculate/count the number of possible combinations for each possible number of marbles. We might also rely on the symmetry between Mel and Nora to cut that workload almost in half (calculate for Mel getting zero through two, multiply by \(2\) to account for Nora getting zero through two, then add the case of Mel and Nora each getting three). But even then, we’d be expending a fair amount of brute force effort.

If we tell the story a bit differently, though, we can make the math a lot better. Rather than focus on one of the people and their number of marbles in the aggregate, focus on one of the colors and the number of that type that one person gets. When it comes to

red marbles, Mel could wind up with

none,

one,

two, or

three – that’s

\(4\) possibilities. As for

green marbles, Mel could have

none,

one, or

two –

\(3\) possibilities. And the

blue marble is a

take-it-or-leave-it proposition –

\(2\) possibilities. (We don’t have to calculate anything for Nora; she’ll just get whatever is left.) Since we’re distributing

red,

blue, and

green, we multiply these results to find out that there are

4∗

3∗

2=\(24\) total options.

The correct answer is \(C\).

Did you notice this problem’s connection to the Unique Factors Trick? What we’re doing here is actually a perfect analogy – we’re taking some quantities of identical items from each of several categories, and it really makes no difference whether those items are various colored marbles or various prime factors. It makes sense that the formula is the same – the number of items in each category plus one (to cover the “choose zero” possibility), and then multiply the results.

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