There are 8 steps to the top of the stairs. In how many ways can we get to the top if we can climb one step or two steps?

There are 8 steps to the top of the stairs. In how many ways can we get to the top if we can climb one step or two steps?

Question:

There are 8 steps to the top of the stairs. In how many ways can we get to the top if we can climb one step or two steps?

Answer:

Let 1 denotes one step and 2 denotes two steps.
Since, either he can climb one step or two steps only, we have to partition 8 steps as a combination of 1 and 2.
The possible combinations are found by choosing number of 2 steps out of all steps as follows:
1) 1, 1, 1, 1, 1, 1, 1, 1
This can be done in $^8C_0=1$ ways.
2) 1, 1, 1, 1, 1, 1, 2
This can be done in $^7C_1=7$ ways.
3) 1, 1, 1, 1, 2, 2
This can be done in $^6C_2=15$ ways.
4) 1, 1, 2, 2, 2
This can be done in $^5C_3=10$ ways.
5) 2, 2, 2, 2
This can be done in $^4C_4=1$ ways.
So, the total number of ways we can get on the top is $1+7+15+10+1=34$.

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