HMMT 二月 2014 · 冲刺赛 · 第 20 题
HMMT February 2014 — Guts Round — Problem 20
题目详情
英文原题
- [ 11 ] A deck of 8056 cards has 2014 ranks numbered 1–2014. Each rank has four suits—hearts,
diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rankand the same suit. How many subsets of the set of cards in this deck have cards from an odd numberof distinct ranks?
HMMT 2014, 22 FEBRUARY 2014 — GUTS ROUND
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解析
英文解析
- [ 11 ] A deck of 8056 cards has 2014 ranks numbered 1–2014. Each rank has four suits—hearts,
diamonds, clubs, and spades. Each card has a rank and a suit, and no two cards have the same rankand the same suit. How many subsets of the set of cards in this deck have cards from an odd numberof distinct ranks?
( )
2014
1 2014 2014
Answer: (16 − 14 ) There are ways to pick k ranks, and 15 ways to pick the suits
2 kin each rank (because there are 16 subsets of suits, and we must exclude the empty one). We therefore
( ) ( ) ( )
2014 2014 2014
1 3 2013
want to evaluate the sum 15 + 15 + · · · + 15 .
1 3 2013
( ) ( ) ( )
2014 2014 2014
2014 1 2 2013 2014 2014
Note that (1 + 15) = 1 + 15 + 15 + . . . + 15 + 15 and (1 − 15) =
1 2 2013
( ) ( ) ( ) 2014 2014
(1+15) − (1 − 15)
2014 2014 2014
1 2 2013 2014
1 − 15 + 15 − . . . − 15 + 15 , so our sum is simply =
1 2 2013 2
1 2014 2014
(16 − 14 ). 2