\(\text{Diane has one 1-cent stamp, two identical 2-cent stamps, and so on, up to nine identical 9-cent stamps. In how many different arrangements can Diane paste exactly 10 cents worth of postage in a row across the top of an envelope? (Note, however, that simply rotating or inverting a stamp, or exchanging the positions of two stamps with the same denomination should be considered the same arrangement.) }\)
Here is my attempt at this. I come up with 16 combinations as follows:
1- 9 + 1 = 10
2 - 8 + 2 = 10
3 - 7 + 3 = 10
4 - 6 + 4 = 10
5 - 5 + 5 = 10
6 - 7 + 2 + 1 = 10
7 - 6 + 3 + 1 = 10
8 - 6 + 2 + 2 = 10
9 - 5 + 4 + 1 = 10
10 - 5 + 3 + 2 = 10
11 - 4 + 4 + 2 = 10
12 - 4 + 3 + 3 = 10
13 - 5 + 2 + 2 + 1 = 10
14 - 4 + 3 + 2 + 1 = 10
15 - 3 + 3 + 3 + 1 = 10
16 - 3 + 3 + 2 + 2 = 10
Only one of your equations is equal to 10.
1- 9 + 1 = -7
2 - 8 + 2 = -4
3 - 7 + 3 = -1
4 - 6 + 4 = 2
5 - 5 + 5 = 5
6 - 7 + 2 + 1 = 2
7 - 6 + 3 + 1 = 5
8 - 6 + 2 + 2 = 6
9 - 5 + 4 + 1 = 9
10 - 5 + 3 + 2 = 10
11 - 4 + 4 + 2 = 13
12 - 4 + 3 + 3 = 14
13 - 5 + 2 + 2 + 1 = 13
14 - 4 + 3 + 2 + 1 = 16
15 - 3 + 3 + 3 + 1 = 19
16 - 3 + 3 + 2 + 2=20
