(1, 1, 8) is a triple of natural numbers which has a sum of 10. Consider (1, 8, 1) and (8, 1, 1) to be the same triple as (1, 1, 8). How many different triples of natural numbers have a sum of 10? Include (1, 1, 8) as one of your triples.
Is there a fast way to do this problem is what I would like to know.
If so, what is it?
To solve this problem efficiently, you can use a technique called "stars and bars" or "balls and urns." This method allows you to distribute a fixed sum among a fixed number of parts, in this case, three parts.
Here's how you can use the stars and bars method to find the number of different triples of natural numbers with a sum of 10:
1. Start by subtracting 1 from the sum (10 - 1 = 9). This adjustment is necessary because the problem specifies "natural numbers," which typically start from 1.
2. Represent the sum as stars (*). In this case, you have 9 stars (*********).
3. Add two bars (|) to divide the stars into three parts. The bars represent the boundaries between the triple's components. Each star to the left of the first bar represents the first component, between the two bars represents the second component, and to the right of the second bar represents the third component.
For example, if the stars and bars are represented as ***|****|**, then the triple would be (3, 4, 2).
4. Now, count the number of ways you can arrange the stars and bars. This can be computed using a combinatorial formula.
The number of ways to arrange n stars and k bars is given by the formula: C(n + k, k), which represents the binomial coefficient or "n choose k."
Therefore, in this case, the number of different triples is C(9 + 2, 2) = C(11, 2).
Simplifying this expression:
C(11, 2) = 11! / (2! * 9!) = (11 * 10) / (2 * 1) = 55.
So the fast way to solve the problem is to use the stars and bars technique and compute the binomial coefficient C(n + k, k), where n is the adjusted sum of 9 and k is 2 (since we have two bars dividing the stars).
Therefore, there are 55 different triples of natural numbers that have a sum of 10, including (1, 1, 8).