Question

在某种纸牌游戏中，一名玩家从 $52$ 张互不相同的牌组成的牌堆中拿到 $10$ 张牌。可以发给该玩家的不同无序手牌数可写成 $158A00A4AA0$ 。数字 $A$ 是多少？

In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0.$ What is the digit $A?$

$2$

$3$

$4$

$6$

$7$

答案：A

知识点：组合数字模运算

难度评级：1620

视频讲解：

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文字解答：

不同手牌数为 $158A00A4AA0$ $10$ ， $A$ 。 $\begin{aligned} \binom{52}{10} &=10\cdot17\cdot13\cdot7\\ &\quad\cdot47\cdot46\cdot11\cdot43. \end{aligned}$

提出末尾的因子十后，得到为求个位数字，只需将这个乘积对 $10$ 取模：由于，且，所以 $A=2$ 。所以正确答案是 A。 $\begin{aligned} A&\equiv7\cdot3\cdot7\cdot7\cdot6\cdot1\cdot3\\ &\equiv2\pmod{10}. \end{aligned}$

After canceling factors in the binomial coefficient, $\begin{aligned} \binom{52}{10} &=10\cdot17\cdot13\cdot7\\ &\quad\cdot47\cdot46\cdot11\cdot43. \end{aligned}$ Dividing $158A00A4AA0$ by the final factor $10,$ the units digit of the remaining product is $A.$

Working modulo $10,$ $\begin{aligned} A&\equiv7\cdot3\cdot7\cdot7\cdot6\cdot1\cdot3\\ &\equiv2\pmod{10}. \end{aligned}$ Hence $A=2.$ Thus, A is the correct answer.

2020 AMC 10B 第 19 题

答案：A

难度评级：1620

视频讲解：

文字解答：

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