Question

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 $A?$

$2$

$3$

$4$

$6$

$7$

Answer: A

Concepts:combinations digits modular arithmetic

Difficulty rating: 1620

Video solution:

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Written solution:

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 Problem 19

Answer: A

Difficulty rating: 1620

Video solution:

Written solution:

Problem 19 in Other Years