A Staircase Permutation Problem

Question:

“Liam’s house has a staircase with 12 steps. He can go down the steps one at a time or two at a time. For example, he could go down 1 step, then 2 steps, then 2, 2, 1, 1, 1, 1, 1. In how many different ways can Liam go down the 12 steps, taking 1 or 2 steps at a time?”

Solution

We break the cases according to the number of “2 steps” (and hence determines the number of “1 step”s in order to get a total of twelve steps between them all). We call each of these counting a “Way” and then proceed to count the “no.of permutations” for each “Way” before adding all of them up to obtain the answer.

(i) Six “2 steps” can be done in 1 way only.

(ii) Five “2 steps” & Two “1 step” in 7!/(5!2!) = 21 ways.

(iii) Four “2 steps” & Four “1 step” in 8!/(4!4!) = 70 ways.

(iv) Three “2 steps” & six “1 step” in 9!/(3!6!) = 84 ways.

(v) Two “2 steps” & “eight “1 step” in 10!/(2!8!) =  45 ways.

(vi) One “2 steps” & ten “1 step” in 11!/(1!10!) = 11 ways.

(vii) No “2 steps” and twelve “1 step” in 1 way only too.

Hence,

Answer = 1 + 21 + 70 + 84 + 45 + 11 + 1 = 233 ways.

That is, Answer = 233 ways.