An elf has a staircase with 10 steps to climb. Each step it takes can cover either one stair or two stairs. In how many ways can the elf climb the stairs? For n stairs?

To find the number of ways the elf can climb the stairs, we can use the concept of dynamic programming.

Let's define a function F(n) that represents the number of ways the elf can climb n stairs. We can break down the problem into smaller subproblems:

- If there are 0 stairs, there is only one way to climb them (by not taking any steps).
- If there is 1 stair, there is only one way to climb it (by taking one step of size 1).
- If there are 2 stairs, there are two ways to climb them (by taking two steps of size 1 or one step of size 2).

For n stairs, the elf can take either one step of size 1 or two steps of size 2.

The number of ways to climb n stairs can be determined by considering the number of ways to climb (n-1) stairs (by taking one step of size 1 at the end) and the number of ways to climb (n-2) stairs (by taking two steps of size 2 at the end). Therefore, we have the recurrence relation:

F(n) = F(n-1) + F(n-2)

Using this recurrence relation, we can compute the number of ways to climb the stairs iteratively:

1. Initialize F(0) = 1 and F(1) = 1.
2. For i = 2 to n, compute F(i) using the recurrence relation F(i) = F(i-1) + F(i-2).
3. The final result will be F(n).

Let's calculate the number of ways the elf can climb the 10 stairs:

F(0) = 1
F(1) = 1

F(2) = F(1) + F(0) = 1 + 1 = 2
F(3) = F(2) + F(1) = 2 + 1 = 3
F(4) = F(3) + F(2) = 3 + 2 = 5
F(5) = F(4) + F(3) = 5 + 3 = 8
F(6) = F(5) + F(4) = 8 + 5 = 13
F(7) = F(6) + F(5) = 13 + 8 = 21
F(8) = F(7) + F(6) = 21 + 13 = 34
F(9) = F(8) + F(7) = 34 + 21 = 55
F(10) = F(9) + F(8) = 55 + 34 = 89

Therefore, there are 89 ways for the elf to climb 10 stairs.

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