different ways of rabbit hopping up flight of 10 steps

[melimarti12]

Laura is training her white rabbit, Ghost, to climb a flight of 10 steps. Ghost can only hop up 1 or 2 steps each time. He never hops down, only up. How many different ways can Ghost hop up the flight of 10 steps?

 

[soroban]

Hello, melimarti12!

I had to make a "list" . . .

Laura is training her white rabbit, Ghost, to climb a flight of 10 steps.
Ghost can only hop up 1 or 2 steps each time. He never hops down, only up.
How many different ways can Ghost hop up the flight of 10 steps?

Ghost climbs the stairs in one-step and two-step hops.


. . \(\displaystyle \begin{array}{cc}\text{Combination} & \text{Number of ways} \\ \hline \\[-3mm] \text{ten 1's} & 1 \\ \text{eight 1's, one 2} & {9\choose8,1} \:=\:9 \\ \\[-3mm] \text{six 1's, two 2's} & {8\choose6,2} \:=\:28 \\ \\[-3mm] \text{four 1's, three 2's} & {7\choose4,3} \:=\:35 \\ \\[-3mm] \text{two 1's, four 2's} & {6\choose2,4} \:=\:15 \\ \text{five 2's} & 1 \\ \hline \end{array}\)


\(\displaystyle \text{Therefore, there are: }\:1 + 9 + 28 + 35 + 15 + 1 \;=\;\boxed{89\text{ ways}}\)

 

[galactus]

I was thinking we could use \(\displaystyle P(10,2)-1\), but is that logically sound?.

We are choosing 2 items out of 10 where order matters. Then, subtracting 1 for some reason.

Does that make sense in some fashion or is it just coincidental?.