how many ways can you get a sum of 5 by adding the numbers 1 and 2? For example:

Question

how many ways can you get a sum of 5 by adding the numbers 1 and 2? For example: 5 = 1 + 1 + 1 + 1 + 1, 5 = 1 + 2 + 1 + 1, ...

I tried brute forcing this question and came up with 7 ways:
1 + 1 + 1 + 1 + 1

1 + 2 + 2

2 + 2 + 1

2 + 1 + 1 + 1

1 + 2 + 1 + 1

1 + 1 + 2 + 1

1 + 1 + 1 + 2

... I can't think of an 8th?!?

I already know that the answer is 8, but there must be some other way to get it, or i just can't think of the last combination. Is there a way to come up with this number by using permutations/combinations or other discrete methods?

Any help would be appreciated, thanks.

Explanation / Answer

You missed '2+1+2'

r we caan have five 1s or 'three 1s and one 2" or two 2's and one 1"

for 1st case: there is only one way

second case: no. of ways = ( 4!/3!) = 4

third case : no . of ways = ( 3! /2!) = 3

total ways = 4+3+1 =8

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