# Math Help - Using Recursion Equations

1. ## Using Recursion Equations

Can someone help me on this problem?

How many vectors (x1,x2......xk) are there for which each xi is a positive
integer such that 1<= xi<=n and x1<x2<.......<xk?

2. If $k=n$, the answer is 1: $(1,2,...,n)$
If $k>n$, the answer is 0.
If $k, we're actually looking at how many k-lengthed increasing sequences there are with entries in $\{1,2,...,n\}$
You should note that for any $k$ elements that we choose, there may only be one such sequence. Also, you can choose any $k$ elements and then simply set them in order. This gives us that there are $\binom{n}{k}$ options.