2008 AMC 10B Problem 1

Below is the professionally curated solution for Problem 1 of the 2008 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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Concepts:basic counting systematic listing

Difficulty rating: 720

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:

If $k$ of the baskets are worth $3$ points and the rest worth $2,$ the total is $2(5-k)+3k=10+k.$

As $k$ ranges over $0,1,\ldots,5,$ the total takes every integer value from $10$ to $15,$ giving $6$ possibilities.

Thus, the correct answer is E.

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