2008 AMC 12B Problem 1

A basketball player made

5

baskets during a game. Each basket was worth either

2

3

points. How many different numbers could represent the total points scored by the player?

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Solution

The total ranges from

5 \cdot 2 = 10

(all two-pointers) to

5 \cdot 3 = 15

(all three-pointers). Swapping one two-pointer for a three-pointer raises the total by exactly

1,

so every integer in between occurs. The possible totals are

10, 11, 12, 13, 14, 15,

which is

6

values. Thus, the correct answer is E.