Combinations: adding 1-5, using only 5 numbers, to get 15 or less

MelomaniacJEFF

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How many ways can you add the numbers 1-5, USING ONLY 5 NUMBERS, to get a total of 15 or less. You may use numbers more than once
1,1,1,1,1 is a possibility since it is less than 15. Remember, 1,2,1,1,1 is a different outcome than 1,1,2,1,1. ORDER DOES MATTER. I know there are over 3,000 Different possibilities, but can you list them all or attach a document. Or could you please explain how I can go about finding them all. Thanks a lot.


 
If you, for whatever reason, really want to list every possibility, the only way I know of is to just write them all out. Of course, this task can be dramatically sped up by using Excel or writing a small script. However, I fail to see how that would help you to solve this problem, as you're not asked to list all possible solutions, only to tell how many there are. And in this specific instance, you'd most likely be better off by finding how many combinations add up to 16 or more, then subtracting that from the total number of combinations.

One way you might try to tackle this problem is to "lock in" some of the numbers and then work with combinations of the remaining numbers. For instance, assume that the first and second values are both 1. Then, how many ways can you arrange the remaining three numbers such that their sum is 14 (that is, 16 - 1 - 1) or more? Then, perhaps go back and "lock in" the second value as 2, and proceed from there.
 
How many ways can you add the numbers 1-5, USING ONLY 5 NUMBERS, to get a total of 15 or less. You may use numbers more than once 1,1,1,1,1 is a possibility since it is less than 15. Remember, 1,2,1,1,1 is a different outcome than 1,1,2,1,1. ORDER DOES MATTER. I know there are over 3,000 Different possibilities, but can you list them all or attach a document. Or could you please explain how I can go about finding them all. Thanks a lot
Below is the expansion of \(\displaystyle (x+x^2+x^3+x^4+x^5)^5\).
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Note that the coefficient of \(\displaystyle x^5\) is one. This is but one way to get the sum of five: \(\displaystyle 1+1+1+1+1+1\)
Note that the coefficient of \(\displaystyle x^6\) is six There are six ways to get the sum of six: \(\displaystyle 2+1+1+1+1+1\)
Note that the coefficient of \(\displaystyle x^{15}\) is 381. There are 381 ways to get the sum of fifteen.
So add up the coefficients of \(\displaystyle x^{k},~k=5,6,\cdots,14,15.\)
That sum is your answer. This called using generating functions. The bad news is that entire courses are devoted to their study.

If you want more insight into this process do this.
Each set \(\displaystyle 1,1,1,1,3;~1,1,1,2,2\) adds to seven. Are there fifteen ways to rearrange these strings?
Is that the coefficient of \(\displaystyle x^7~?\)
 

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