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Sample Problems: Combinatorics -- SAT, GRE, GMAT, Statistics, Discrete Math


Of the vast amount of math taught in high school, combinatorics is usually the most baffling for students. In my ten years of teaching, I've never had a student who felt totally confident about counting problems -- I myself didn't feel I really understood them until I went to college! -- and the most typical reaction to them is immediate fear or frustration: students often give up as soon as they see one, before they even attempt a solution. Why? Probably because many high school math teachers don't do a good job of explaining the basic concepts with concrete examples; instead, they often present a bunch of formulas to be memorized, without conveying any intuition about when to use the formulas or where they come from.

But you won't earn a stellar score on the SAT, GRE, or GMAT if you can't master the basics of combinatorics. To see if you're up to speed, take a look at the following challenging problems, all based on the following scenario:


Luigi's Pizza Shop offers three types of crust, five types of toppings, and four types of cheese. Suppose a pizza at Luigi's consists of a crust and at least one type of cheese; it may have no toppings, one topping, or more than one topping, as well as more than one cheese. How many different types of pizza can be made:

0. with exactly 5 toppings and 4 cheeses? (Hint: I called this #0 because it's very easy!)

1. with exactly 3 toppings and 2 cheeses?

2. with exactly 2 toppings and 3 cheeses?

3. with exactly 2 toppings and 4 cheeses?

4. with exactly 3 toppings?

5. with exactly 4 toppings?

6. with exactly 4 cheeses?

7. with at least two cheeses?

8. with at most four toppings?

9. with at least two cheeses and at most four toppings?

10. in total?

Extra credit: Can you say which of the above problems were easier than others, and explain what made the harder problems more difficult?


If you work with me, you'll learn how to attack these problems and countless others like them so you feel absolutely confident your answer is correct. We'll start out by drawing very concrete pictures so you understand exactly what's involved in answering a counting problem and where the general solution comes from.

To get the answers and start learning how to ace that next exam, send me an email and let's schedule a meeting today!