- $x_1 + x_2 + x_3 + x_4 + x_5 = 100$
- $x_1 + x_2 + x_3 + x_4 + x_5 \leq 100$
- $x_1 + x_2 + x_3 + x_4 + x_5 < 100$ with all $x_i > 0$
For the first one I said that the answer was $104C4$ and for the second one I said the answer was $105C5$. Are these two correct? If not, how do I do it? As for question 3, I have no idea how to do it. Any help would be appreciated.
Best Answer
The first two are correct, assuming you are counting things like $x_1 = 60, x_2 = 40$ as distinct from $x_1 = 40, x_2 = 60$. The idea here is that you're spreading a certain number of balls into a certain number of bins. The equation you used is usually derived via a stars and bars argument.
For part (a), you're basically distributing 100 balls into 5 bins, so the solution would be $104 \choose 4$. For part (b), you can imagine having 6 bins instead; the 100 balls are distributed between them, and the number of balls in the first five is then less than or equal to 100. The answer is therefore $105 \choose 5$.
Since this looks like a homework problem, I won't give you the straight answer to part (c), but think of it this way: if I handed you 100 balls and told you to put them into 5 bins such that there's at least one ball in each bin, what's the first thing you would do? What would you do after that?