Question

How many vectors $x_1,...,x_k$are there for which each role="math" localid="1647853392605" $x_i$is a positive integer such that role="math" localid="1647853435585" $1\le x_i\le n$androle="math" localid="1647853511159" $x_1<x_2<···<x_k$?

Short answer

The number of vectors are${}^{n}C_k=\frac{n!}{k!(n-k)!}$.

Step-by-step solution

Step 1. Given information.

It is given that,

$x_i$is a positive integer.

$1\le x_i\le n$, it means all the numbers in the set lies in the range $(1,n)$.

$x_1<x_2<···<x_k$, it means the numbers should be ascending order. As out of $n$, $k$ distinct integers are chosen so there can be only one way of arrangement.

Step 2. State the answer.

So, to get a set of numbers fulfilling the given conditions is same as selecting $k$ numbers randomly from $n$ numbers, which can be done in ${}^{n}C_k=\frac{n!}{k!(n-k)!}$ways.