Q. 1.6 - A First Course in Probability

Question

How many vectors $x_1, . . . , x_k$ are there for which each $x_i$ is a positive integer such that $1 \le x_i\le n$ and  $x_1 < x_2 < · · · < x_k$?

Step-by-Step Solution

Answer

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

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.