HMMT 十一月 2021 · 冲刺赛 · 第 12 题

HMMT November 2021 — Guts Round — Problem 12

[8] Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The m probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as , n where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2021, November 13, 2021 — GUTS ROUND Organization Team Team ID#

解析

[8] Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as m , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n Proposed by: Zhao Yu Ma Answer: 26597 ( ) 3 3 27 Solution: The probability that all three cards drawn are face cards is = . In that case, 13 2197 the sum is 30 and therefore a multiple of 10. Otherwise, one of the cards is not a face card, so its point value p is drawn uniformly from values from 1 to 10. The sum of the values of the other two cards uniquely determines the point value p for which the sum of all three values is a multiple of 10. ( ) 27 1 27 244 Therefore, the total probability is + 1 − = . 2197 10 2197 2197