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