AMC12 2008 B

AMC12 2008 B · Q1

AMC12 2008 B · Q1. It mainly tests Fractions, Basic counting (rules of product/sum).

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?

一名篮球运动员在一场比赛中投中了 $5$ 个篮。每个篮得分要么是 $2$ 分，要么是 $3$ 分。该运动员的总得分可能有多少种不同的数值？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 6 6

Answer

Correct choice: (E)

正确答案：（E）

Solution

If the basketball player makes $x$ three-point shots and $5-x$ two-point shots, he scores $3x+2(5-x)=10+x$ points. Clearly every value of $x$ yields a different number of total points. Since he can make any number of three-point shots between $0$ and $5$ inclusive, the number of different point totals is $6 \Rightarrow E$. Stars and bars can also be utilized to solve this problem. Since we need to decide what number of 2's and 3's are scored, and there are a total of 5 shots. It can be written like such: _ _ _ | _ _. Solving this, we get $6 \Rightarrow E$.

如果该篮球运动员投中了 $x$ 个三分球和 $5-x$ 个两分球，那么他得到的总分为 $3x+2(5-x)=10+x$ 分。显然，每个不同的 $x$ 都会得到不同的总得分。由于他投中的三分球个数可以是从 $0$ 到 $5$（含端点）的任意整数，因此不同的总得分个数为 $6 \Rightarrow E$。也可以用隔板法（stars and bars）来解。因为我们需要决定得了多少个 2 分和 3 分，并且总共投中 5 个球。可以写成：_ _ _ | _ _。解得共有 $6 \Rightarrow E$。

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