Q. 1.14 - A First Course in Probability

Question

Determine the number of vectors $(x_1, . . . , x_n)$ such that each $x_i$ is a positive integer and

$\sum _{i=1}^n{x}_i\le k$

where $k\ge n$.

Step-by-Step Solution

Answer

The no. of vectors are $\sum _{j=n}^k\left(\begin{array}{c}j-1\\ n-1\end{array}\right)$.

Step 1. Given information.

It is given that $\sum _{i=1}^n{x}_i\le k$

and $k\ge n$.

Step 2. Find the no. of vectors.

There are $\left(\begin{array}{c}j-1\\ n-1\end{array}\right)$ positive vectors whose sum is $j$.

So, the possible no. of vectors are $\sum _{j=n}^k\left(\begin{array}{c}j-1\\ n-1\end{array}\right)$.