# Thread: Combinatorics and Permuation Question

1. ## Combinatorics and Permuation Question

I was looking at some problems in a book asking:

1. How many ways are there to divide a 331 page book into 8 chapters assuming each chapter is an integer number of pages?

2. Prove that for all n greater than 2, C(n,3) = C(2,2) + C(3,2) + ... + C(n-1,2).

I've looked at them for awhile but not really sure how to approach it. Any ideas?

Thanks

2. Originally Posted by Poplock
1. How many ways are there to divide a 331 page book into 8 chapters assuming each chapter is an integer number of pages?
You are asking to solve,
$\displaystyle x_1+x_2+...+x_8 = 331$ where $\displaystyle x_i \geq 1$.
The number of solutions is $\displaystyle {{330}\choose 7}$.
2. Prove that for all n greater than 2, C(n,3) = C(2,2) + C(3,2) + ... + C(n-1,2)
$\displaystyle {k\choose 2} = \frac{1}{2}k^2 - \frac{1}{2}k$

You want to show $\displaystyle \sum_{k=2}^{n-1} \frac{1}{2}k^2 - \frac{1}{2}k = {n\choose 3} = \frac{n(n-1)(n-2)}{6}$

Here use the identities, $\displaystyle \sum_{k=1}^n k = \frac{k(k+1)}{2}$ and $\displaystyle \sum_{k=1}^n k^2 = \frac{k(k+1)(2k+1)}{6}$.