How many wasy to make 10 out of 1, 2 and 3?

[TheIceBerg]

Hello,

Imagine I have a pile of bills in the following denominations: $1, $2 and $3. How many ways can I make exactly $10 using these bills? Let's assume I have more than enough of each bill to make $10. I have come up with 14 ways just by listing them, but want to make sure I am not missing any. If the raw mathematics of it are digestible to a layperson, the "shortcut" would be appreciated as well in case I need to do this again with different values. Here are the solutions I have found so far:

10 $1
8 $1 and 1 $2
7 $1 and 1 $3
6 $1 and 2 $2
5 $1, 1 $2 and 1 $3
4 $1 and 3 $2
4 $1 and 2 $3
3 $1, 2 $2 and 1 $3
2 $1 and 4 $2
2 $1, 1 $2 and 2 $3
1 $1, 3 $2 and 1 $3
1 $1 and 3 $3
5 $2
2 $2 and 2 $3

Is that all of the possibilities? Is there a quick way to check? Any help would be appreciated.

 

[Dr.Peterson]

Hello,

Imagine I have a pile of bills in the following denominations: $1, $2 and $3. How many ways can I make exactly $10 using these bills? Let's assume I have more than enough of each bill to make $10. I have come up with 14 ways just by listing them, but want to make sure I am not missing any. If the raw mathematics of it are digestible to a layperson, the "shortcut" would be appreciated as well in case I need to do this again with different values. Here are the solutions I have found so far:

10 $1
8 $1 and 1 $2
7 $1 and 1 $3
6 $1 and 2 $2
5 $1, 1 $2 and 1 $3
4 $1 and 3 $2
4 $1 and 2 $3
3 $1, 2 $2 and 1 $3
2 $1 and 4 $2
2 $1, 1 $2 and 2 $3
1 $1, 3 $2 and 1 $3
1 $1 and 3 $3
5 $2
2 $2 and 2 $3

Is that all of the possibilities? Is there a quick way to check? Any help would be appreciated.

It looks like you got them all.

I would start with the largest number of the largest denomination, and work down, like this:

[FONT=courier new]3 2 1
- - -
3 0 1
2 2 0
2 1 2
2 0 2
...[/FONT]

Do you see how easy it is to be sure you get all of them, just because of the orderliness of the method?

It's also possible to recognize a pattern and develop a sort of formula to count, so that it may not be necessary to actually make the whole list; but that depends heavily on the particular numbers involved, and I am not aware of a general formula or shortcut.

I get 14 ways, which match up with yours. Looking closely, I see that you followed the opposite strategy to mine, which is fine, though possibly a little harder to see patterns.

 

[pka]

Imagine I have a pile of bills in the following denominations: $1, $2 and $3. How many ways can I make exactly $10 using these bills? Let's assume I have more than enough of each bill to make $10. I have come up with 14 ways just by listing them, but want to make sure I am not missing any. If the raw mathematics of it are digestible to a layperson, the "shortcut" would be appreciated as well in case I need to do this again with different values. Here are the solutions I have found so far:
10 $1
8 $1 and 1 $2
7 $1 and 1 $3
6 $1 and 2 $2
5 $1, 1 $2 and 1 $3
4 $1 and 3 $2
4 $1 and 2 $3
3 $1, 2 $2 and 1 $3
2 $1 and 4 $2
2 $1, 1 $2 and 2 $3
1 $1, 3 $2 and 1 $3
1 $1 and 3 $3
5 $2
2 $2 and 2 $3
Is that all of the possibilities? Is there a quick way to check? Any help would be appreciated.

The good news is that fourteen is the correct answer.
This is a generating function for this question: \(\displaystyle (\sum\limits_{k = 0}^{10} {{x^k}} )(\sum\limits_{k = 0}^5 {{x^{2k}}} )(\sum\limits_{k = 0}^3 {{x^{3k}}} )\)
You can see its expansion HERE. You will see the term \(\displaystyle 14x^{10}\). The coefficient \(\displaystyle 14\) of \(\displaystyle x^{10}\) answers your question.
Ask yourself why that product works? If this interests you there is a delightful paperback book Mathematics of Choice by Ivan Niven.