2008 AMC 12B Problem 1

Question

A basketball player made

5

baskets during a game. Each basket was worth either

2

or

3

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

2

3

4

5

6

The total is smallest with all

2

-point baskets and largest with all

3

-point baskets

Every integer between

5 \cdot 2

and

5 \cdot 3

is attainable

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.

Problem 1 in other years