How many combinations to make a sum?

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SUMMARY

The discussion focuses on determining the number of combinations of tokens valued at 1, 5, and 10 that sum to 17, represented mathematically by the equation 10x + 5y + z = 17, where x, y, and z are non-negative integers. Participants highlight that while listing combinations is a straightforward approach, leveraging arithmetic knowledge can yield more efficient solutions. The problem is framed within the context of finding integer points in the positive octant of a plane defined by the equation.

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fk378
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Hi all,
Saw this problem and was wondering if there was a simpler way to do this besides listing out the possible combinations.

In a game, each token has one possible value: 1, 5, or 10. How many different combinations of these tokens will give us a total sum of 17?
 
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You need to rewrite the question mathematically then see if you can rework it into something that looks solvable. i.e.

Find all points (x,y,z) which satisfy 10x+5y+z=17 - where x,y,z are integers greater than or equal to zero.

The equation is of a plane... so the question is looking for the number of integer points in the plane that is in the positive octant.

It is usually easier to just list and count.
You can use your knowledge of arithmetic to find shortcuts though.
 

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