Question

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

role="math" localid="1649161391681" $\sum _{i=1}^n{x}_i\le k$

where$k\ge n$.

Short answer

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

Step-by-step solution

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$\sum _{j=n}^k\left(\begin{array}{c}j-1\\ n-1\end{array}\right)$.