There are three distinct ways to add four positive odd numbers to obtain 10:

Question

1 + 1 + 3 + 5 = 10
1 + 1 + 1 + 7 = 10
1 + 3 + 3 + 3 = 10
Here, distinct means that changing the order of the numbers on the left-hand side of an
equation does not count as a new solution. In how many distinct ways can we add eight odd
positive numbers to obtain 20? Justify your response. Hint: be systematic in your approach!

Answer 1

Perhaps there is a more elegant way, but the following should work.

Start with smallest to largest numbers:
1+1+1+1+1+1+1+13=20
1+1+1+1+1+1+3+11=20
1+1+1+1+1+1+5+9=20
1+1+1+1+1+1+7+7=20
1+1+1+1+1+3+3+9=20
1+1+1+1+1+3+5+7=20
1+1+1+1+1+5+5+5=20
1+1+1+1+3+....

Answer 2

keep going......