2008 AMC 12B Problem 1

Below is the professionally curated solution for Problem 1 of the 2008 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12B solutions, or check the answer key.

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Concepts:counting integers in a range

Difficulty rating: 920

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?

$2$

$3$

$4$

$5$

$6$

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.

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