SOLUTION: Nita, a six-year-old, can stretch her legs to take 2 stairs at one time. How many different ways can she climb 6 stairs using any combination of 1 or 2 stairs?

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Question 239281: Nita, a six-year-old, can stretch her legs to take 2 stairs at one time. How many different ways can she climb 6 stairs using any combination of 1 or 2 stairs?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
She can do 6 steps one at a time only 1 way
She can do 4 steps one at a time and 1 step 2 at a time 5 ways
21111
12111
11211
11121
11112
She can do 2 steps one at a time and 2 steps 2 at a time 6 ways
1122
1212
1221
2112
2121
2211
She can do 3 steps 2 at a time only 1 way
222

Total number of ways equals 1 + 5 + 6 + 1 = 13

The formula is the number of possible combinations without taking order into account.

0 steps 2 at a time and 6 steps 1 at a time winds up being 6! / 0!6! = 1/1 = 1

1 step 2 at a time and 4 steps 1 at a time winds up being 5! / 1!4! = 5/1 = 5

2 steps 2 at a time and 2 steps 1 at a time winds up being 4! / 2!2! = 12/2 = 6

3 steps 3 at a time and 0 steps 1 at a time winds up being 3! / 3!0! = 1

The combination formula is (n!) / (x! * (n-x)!)

Each time she does 2 steps at a time, the total number of steps she has to take is reduced by 1.
6 steps one at a time and 0 steps 2 at a time becomes a total of 6 steps that she has to take.
4 steps one at a time and 1 step 2 at a time becomes a total of 5 steps that she has to take.
2 steps one at a time and 2 steps 2 at a time becomes a total of 4 steps that she has to take.
0 steps one at a time and 3 steps 2 at a time becomes a total of 3 steps that she has to take.

It's hard to see.

You have to work it out and then the pattern becomes apparent.