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2020 AMC 10B Problem 19

Problem 19 of 25HarderNumber TheoryCounting & Probability

In a certain card game, a player is dealt a hand of 1010 cards from a deck of 5252 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as 158A00A4AA0.158A00A4AA0. What is the digit A?A?

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Solution

After canceling factors in the binomial coefficient, (5210)=10⋅17⋅13⋅7⋅47⋅46⋅11⋅43. \begin{aligned} \binom{52}{10} &=10\cdot17\cdot13\cdot7\\ &\quad\cdot47\cdot46\cdot11\cdot43. \end{aligned} Dividing 158A00A4AA0158A00A4AA0 by the final factor 10,10, the units digit of the remaining product is A.A. Working modulo 10,10, A≡7⋅3⋅7⋅7⋅6⋅1⋅3≡2(mod10). \begin{aligned} A&\equiv7\cdot3\cdot7\cdot7\cdot6\cdot1\cdot3\\ &\equiv2\pmod{10}. \end{aligned} Hence A=2.A=2. Thus, A is the correct answer.

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Concepts: combinations · digits · modular arithmetic

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.