Luke the cat

Yay Im a cat~ Anyways to find the number of ways: Number of ways when there are 6 "2"s and 0 "1"s: 6!/6!=1 Number of ways when there are 5 "2"s and 2 "1"s: (5+2)!/(5!x2!)=21 Number of ways when there are 4 "2"s and 4 "1"s: (4+4)!/(4!x4!)=70 ............. Number of ways when there are 0 "2"s and 12 "1"s: 12!/12!=1 Therefore the total number of ways: 1+21+70+84+45+11+1=233

It's actually simple to calculate it by recurrence. If we call $f(n)$ the number of ways the cat can get the bowl, and since there's just $2$ different ways of it starts doing it, we have:

i) Either the cat jumps $1$ step at a time, reaching the first step; and the cat has now $f(n-1)$ ways of getting the bowl.

ii) Or the cat jumps $2$ steps at a time, reaching the second step; and the cat has now $f(n-2)$ ways of getting the bowl.

So we have: $f(n)=f(n-1)+f(n-2)$

We can easily calculate it by knowing that $f(1)=1$ (if there's just one step, then there's just one way to gettong the bowl: by jumping just $1$ step) and $f(2)=2$ (if there's just two steps, then there's two ways to getting the bowl: either jumping $1$ step at a time or jumping $2$ steps at a time, reaching the bowl).

Therefore:

$f(1)=1$

$f(2)=2$

$f(3)=1+2=3$

$f(4)=2+3=5$

$f(5)=3+5=8$

$f(6)=5+8=13$

$f(7)=8+13=21$

$f(8)=13+21=34$

$f(9)=21+34=55$

$f(10)=34+55=89$

$f(11)=55+89=144$

$f(12)=89+144=\boxed{233}$