2010
Problem - 1150
Everyday at school, Jo climbs a flight of $6$ stairs. Joe 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?
This is a typical problem that can be tackled using recursion. Let $a_n$ be the number of different ways to climb $n$ stairs. Then there are $3$ possibilities when Jo can reach the $n^{th}$ step depending on the number of stairs in his last climb. So the recursion is $$a_n = a_{n-1}+a_{n-2}+a_{n-3}$$
Meanwhile, it can determine manually that $a_1=1$, $a_2=2$, and $a_3=4$. It follows that $$\begin{array}{rll} a_4=\ &4+2+1&=\ 7\\a_5=\ &7+4+2&=\ 13\\a_6=\ & 13+7+4&=\ \boxed{24}\end{array}$$