HMMT 二月 2012 · 冲刺赛 · 第 12 题

英文解析

12. [ 5 ] Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cardsnumbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round,
    Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a cardwith greater value gets a point. At the end of five rounds, the player with the most points wins. If
    Gammadorf starts with a hand of 1 , 5 , 10 , 15 , 20, how many five-card hands of the fifteen remainingcards can Knot draw which always let Knot win (assuming he plays optimally)?
    Answer: 2982 Knot can only lose if all of his cards are lower than 10; if not he can win by playingthe lowest card that beats Gammadorf’s card, or if this is not possible, his lowest card, each turn.
    ( ) ( ) ( )
    7 15 7
    There are = 21 losing hands, so he has − possible winning hands.
    5 5 5