2008 AMC 12B 第 1 题

Question

一名篮球运动员在一场比赛中投进了

5

个球。每次进球得

2

分或

3

分。该运动员总得分可能有多少种不同的数值？

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

总分最小时全是

2

分球，最大时全是

3

分球。

The total is smallest with all

2

-point baskets and largest with all

3

-point baskets

5 \cdot 2

和

5 \cdot 3

之间的每个整数都能达到。

Every integer between

5 \cdot 2

and

5 \cdot 3

is attainable

总分从

5 \cdot 2 = 10

（全是两分球）到

5 \cdot 3 = 15

（全是三分球）。每把一个两分球换成三分球，总分正好增加

1

，所以中间的每个整数都会出现。

可能的总分是

10, 11, 12, 13, 14, 15

，共有

6

个值。

因此，正确答案是 E。

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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