Exercise 7.9
"How many ways can you find three natural numbers, each bigger than 0, such that their sum is 3000. Suppose that (i)we care about the order of the numbers and (ii) we do not care about the order of the numbers." How would you go about solving this problem? Is there a certain equation or sets of equations that would help? Thank you!
weisbart: Feb. 3, 2015, 2:49 p.m.
Anyone want to try this one? Why is it different from the example using 1000?
neelems: Feb. 4, 2015, 4:10 p.m.
i don't understand how the order of the numbers affects the number of ways
AZobi: Feb. 7, 2015, 12:58 p.m.
For the example using 1000, the subset denoted by C was empty because 1,000 is not divisible by three. In other words, no three equal numbers add up to give us 1,000. 3,000, however, is divisible by three. So the C subset wouldn't be empty.
ArieleAndalon: Feb. 7, 2015, 6:38 p.m.
@neelems: The order of the numbers refers to the order in which we sum them. In this example, let's say that the numbers are 900, 700, and 1400. If the order matters, 900+700+1400 is different from 1400+900+700. So if order matters, the number of ways that we can write 3000 as a sum of the three natural numbers is greater than if order didn't matter. We usually find the subsets assuming that order does matter and then we have to de-order the subsets in the final answer if order does not matter.