HMMT 二月 2015 · COMB 赛 · 第 7 题

英文解析

7. 2015 people sit down at a restaurant. Each person orders a soup with probability . Independently,
   each person orders a salad with probability . What is the probability that the number of people who 12
   ordered a soup is exactly one more than the number of people who ordered a salad?2
   Combinatorics
   4030 4030 4032 4030 − 2
   ( ) ( ) ( ) ( )
   2016 2014 2016 2015
   Answer: OR OR Solution 1. Note that total soups = total salads + 1
   4030 4030 4031
   2 2 2
   ( )
   2015+2015
   is equivalent to total soups + total not-salads = 2016. So there are precisely possibilities,
   2016
   4030
   ( )
   2015+2015 2016
   each occurring with probability (1 / 2) . Thus our answer is .
   4030
   ( )( )2
   ∑
   2014
   2015 2015
   Solution 2. To count the number of possibilities, we can directly evaluate the sum .
   i =0
   i i +1
   ( ) ( ) ( )( )
   ∑
   2014
   2015 2015 2015 2015
   One way is to note = , and finish with Vandermonde’s identity: =
   i =0
   i +1 2014 − i i 2014 − i
   ( ) ( ) ( )
   2015+2015 4030 4030 = (which also equals ).
   2014 2014 2016
   ( ) ( ) ( )( ) ( )
   ∑
   2014
   2015 2015 2015 2015 2015+2015
   (We could have also used = to get = directly, which isi =0
   i 2015 − i 2015 − i i +1 2016
   closer in the spirit of the previous solution.)
   ( )( )
   ∑
   2014
   2015 2015
   Solution 3 (sketch). It’s also possible to get a handle on by squaring Pascal’si =0
   i i +1
   4032 4030
   ( ) ( ) ( ) − 2
   ( ) ( )
   2015 2015 2016
   2016 2015
   identity + = and summing over 0 ≤ i ≤ 2014. This gives an answer of ,
   4031
   i i +1 i +1 2
   ( ) ( )
   4032 4031
   4032
   which can be simplified by noting = , and then applying Pascal’s identity.
   2016 2016 2015