AMC8 2017

AMC8 2017 · Q10

AMC8 2017 · Q10. It mainly tests Combinations, Probability (basic).

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?

一个盒子里有五张卡片，编号 1、2、3、4 和 5。从盒子里不放回地随机抽取三张卡片。4 是所抽最大值的概率是多少？

(A) $\frac{1}{10}$ $\frac{1}{10}$

(B) $\frac{1}{5}$ $\frac{1}{5}$

(D) $\frac{2}{5}$ $\frac{2}{5}$

(E) $\frac{1}{2}$ $\frac{1}{2}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

There are $\binom{5}{3}$ possible groups of cards that can be selected. If $4$ is the largest card selected, then the other two cards must be either $1$, $2$, or $3$, for a total $\binom{3}{2}$ groups of cards. Then, the probability is just ${\frac{{\dbinom{3}{2}}}{{\dbinom{5}{3}}}} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}$.

可能抽取的卡片组有 $\binom{5}{3}$ 种。如果 $4$ 是所抽的最大卡片，则另外两张必须来自 1、2 或 3，总共 $\binom{3}{2}$ 种卡片组。然后，概率就是 ${\frac{{\dbinom{3}{2}}}{{\dbinom{5}{3}}}} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}$。

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