You are climbing a stair case. It takes n steps to reach to the top. Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?

(n^2 + n) / 2 ways

Explanation:

Each time you can climb 1 or 2 steps to the top (n steps)

Number of ways climbing 1 step to the top = n combination 1 = n! / (n-1) 1! = n (n-1) ! / (n-1) ! = n

Number of ways climbing 2 steps to the top = n combination 2 = n! / (n-2) !2! = n (n-1) (n-2) !/2 (n-2) ! = (n^2 - n) / 2

Total number of ways = n + (n^2 - n) / 2 = (2n + n^2 - n) / 2 = (n^2 + n) / 2 ways