A GRE Problem I am confused by

[phiscobisco]

How do I go about answering this? Very confused on permutations and how to know what to put where (specifically on the bottom)?

Question: When making holiday cookies, Marge has a choice of three cookie shapes, four frosting flavors, and seven types of sprinkles. If Marge has to put three cookies in a box, with each cookie having at least one difference from the other two, how many different combinations of cookies can she put in a box?

Explanation: First, figure out how many different types of cookies there are: 3*4*7 = 84.

Next, use the formula for combinations (nCr) to figure out how many combinations there are:

n = 84
r = 3

→ 84!/(r!(n-r)!) = 84!/(6(81!)) = (84*83*82)/6 = 95284

[tommywallach]

Hey Phisco,

First off, this is not a realistic GRE question. The GRE very rarely tests combinatorics, and when they do, they tend to be quite simple. I promise you and whoever is reading this that you are highly unlikely to see a question like this on your GRE.

However, I'll teach it anyway!

Believe it or not, I'm not a fan of the equations. As you can see, they fall apart when you're trying to deal with complex issues. Instead, I like to think everything out logically, with the slots method.

Slots Method:

1) Find # of slots (how many decisions do you get to make)
2) Fill in each slot with number of options at each decision point
3) Find out if Order Matters or Not
4) If order matters, multiply slots together; if order doesn't matter, multiply slots together and divide by the # of slots factorial

Question: When making holiday cookies, Marge has a choice of three cookie shapes, four frosting flavors, and seven types of sprinkles. If Marge has to put three cookies in a box, with each cookie having at least one difference from the other two, how many different combinations of cookies can she put in a box?

As you said, we need first to work out how many different varieties of cookies there are. That's three slots: one each for shape, frosting flavor, and sprinkle type:

__ __ __ ---> 3 * 4 * 7 = 84

Now, we need to put three cookies in the box. For the first cookie, we could choose any of the 84 varieties. For the second cookie, we need at least one difference, which means there are 83 choices (only one cookie in the universe is exactly the same as the first one), and for the third cookie, we have 82 choices (because we can't pick the exact same cookie as we chose either the first or second time).

84 * 83 * 82 --->

We also know that order DOES NOT matter, which means we divide by the # of slots factorial:

(84 * 83 * 82) / 3! = The answer you already gave.

Hope that makes slightly more sense!

-t