AMC8 2005

AMC8 2005 · Q16

AMC8 2005 · Q16. It mainly tests Pigeonhole principle.

A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?

一个五脚火星人有一个抽屉，里面装满了红、白、蓝色的袜子，每种颜色至少有五只袜子。火星人闭眼一次抽出一只袜子。要保证有5只相同颜色的袜子，火星人需要取出多少只袜子？

(A) 6 6

(B) 9 9

(D) 13 13

(E) 15 15

Answer

Correct choice: (D)

正确答案：（D）

Solution

(D) It is possible for the Martian to pull out at most 4 red, 4 white and 4 blue socks without having a matched set. The next sock it pulls out must be red, white or blue, which gives a matched set. So the Martian must select $4 \times 3 + 1 = 13$ socks to be guaranteed a matched set of five socks.

（D）火星人在不形成一组匹配袜子的情况下，最多可以抽出4只红袜、4只白袜和4只蓝袜。接下来再抽出的那只袜子必定是红、白或蓝中的一种，从而形成一组匹配袜子。因此，火星人必须抽取 $4 \times 3 + 1 = 13$ 只袜子，才能保证得到一组由五只袜子组成的匹配集合。

Topics

CP.005 Pigeonhole principle