HMMT 二月 2002 · 冲刺赛 · 第 24 题
HMMT February 2002 — Guts Round — Problem 24
题目详情
英文原题
- [7] A restricted path of length n is a path of length n such that for all i between 1 andn − 2 inclusive, if the i th step is upward, the i + 1 st step must be rightward.
Find the number of restricted paths that start at (0 , 0) and end at (7 , 3). 4
解析
英文解析
- A restricted path of length n is a path of length n such that for all i between 1 andn − 2 inclusive, if the i th step is upward, the i + 1 st step must be rightward.
Find the number of restricted paths that start at (0 , 0) and end at (7 , 3).
Solution: This is equal to the number of lattice paths from (0 , 0) to (7 , 3) that use onlyrightward and diagonal (upward+rightward) steps plus the number of lattice paths from
(0 , 0) to (7 , 2) that use only rightward and diagonal steps, which is equal to the number ofpaths (as defined above) from (0 , 0) to (4 , 3) plus the number of paths from (0 , 0) to (5 , 2),
( ) ( )
4+3 5+2
or + = 56 .
3 2