How many combinations to make a sum?

In summary, the conversation discusses finding a simpler way to determine the number of different combinations of tokens that add up to a total sum of 17 in a game. This can be rewritten mathematically as finding all points (x,y,z) that satisfy 10x+5y+z=17, where x,y,z are integers greater than or equal to zero. It is suggested to approach this problem by using arithmetic shortcuts rather than listing and counting all possible combinations.
  • #1
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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  • #2
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.
 

1. How do you calculate the number of combinations to make a sum?

The number of combinations to make a sum can be calculated using the formula nCr = n! / (r!(n-r)!), where n represents the total number of items and r represents the number of items being chosen. For example, if you have 10 items and need to choose 3, the number of combinations would be 10C3 = 10! / (3!(10-3)!) = 120.

2. What is the difference between combinations and permutations?

Combinations and permutations both involve counting the number of ways to arrange or choose a set of items, but they differ in the order in which the items are arranged. Combinations do not consider the order, whereas permutations do. For example, choosing three items out of a set of four would result in 4 combinations but 24 permutations, as the order of the three chosen items matters in permutations.

3. How can I use combinations to solve real-world problems?

Combinations can be used to solve real-world problems in various fields such as mathematics, statistics, and computer science. For example, in statistics, combinations can be used to calculate the probability of certain events occurring. In computer science, combinations can be used to generate all possible combinations of a set of characters for password creation.

4. What is the significance of the combination formula in mathematics?

The combination formula is significant in mathematics as it allows us to calculate the number of ways to choose a subset of items from a larger set. This is useful in various mathematical concepts and problem-solving, such as in probability, counting principles, and binomial expansions.

5. Are there any limitations to the combination formula?

Yes, there are limitations to the combination formula. It assumes that all items are distinct and that the order in which the items are chosen does not matter. In some cases, this may not be true, and other methods, such as permutations, may need to be used. Additionally, the combination formula may not be applicable if the items being chosen have restrictions or if repetition is allowed.

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