2010 AMC 8 Problem 25

Below is the professionally curated solution for Problem 25 of the 2010 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 8 solutions, or check the answer key.

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Concepts:partitions and compositions recursive counting

Difficulty rating: 1540

25.

Every day at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1,$ $2,$ or $3$ at a time. For example, Jo could climb $3,$ then $1,$ then $2.$ In how many ways can Jo climb the stairs?

$\ 13$

$\ 18$

$\ 20$

$\ 22$

$\ 24$

Solution:

Let $w_n$ be the number of ways to climb $n$ stairs. For $n\ge4$ , the first step can be $1$ , $2$ , or $3$ stairs, so $w_n=w_{n-1}+w_{n-2}+w_{n-3}$ .

We have $w_1=1$ , $w_2=2$ , and $w_3=4$ . Therefore $w_4=4+2+1=7$ , $w_5=7+4+2=13$ , and $w_6=13+7+4=24$ .

Thus, the answer is E .

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