1. Tardigrade
  2. Question
  3. Mathematics
  4. The number of non-negative solutions of x1+x2+x3+, ldots,+xn ≤ n (where n is positive integer) is

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Q. The number of non-negative solutions of $x_{1}+x_{2}+x_{3}+, \ldots,+x_{n} \leq n$ (where $n$ is positive integer) is

Permutations and Combinations

Solution:

In general, we know that
For the distribution equation
$x_{1}+x_{2}+x_{3}+\ldots+x_{n} \leq n$
Let required ways $=W$
$\Rightarrow W=\begin{Bmatrix}\text { No. of ways of } \\ \text { distributing } \\ 1 \text { item }\end{Bmatrix}+\begin{Bmatrix}{c}\text { No. of ways of } \\ \text { distributing } \\ 2 \text { items }\end{Bmatrix}+\ldots+\begin{Bmatrix}\text { No. of Ways of } \\ \text { distributing } \\ n \text { items }\end{Bmatrix}$
$={ }^{1+n-1} C_{n-1}+{ }^{2+n-1} C_{n-1}+\ldots+{ }^{n+n-1} C_{n-1}$
$={ }^{n} C_{n-1}+{ }^{n+1} C_{n-1}+\ldots+{ }^{2 n-1} C_{n-1}$
$=\left({ }^{n} C_{n-1}+{ }^{n} C_{n}\right)+{ }^{n+1} C_{n-1}+\ldots+{ }^{2 n-1} C_{n-1}-{ }^{n} C_{n}$
$=\left\{\left({ }^{n+1} C_{n}+{ }^{n+1} C_{n-1}\right)+\ldots+{ }^{2 n-1} C_{n-1}\right\}-{ }^{n} C_{n}$
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$=\left({ }^{2 n-1} C_{n}+{ }^{2 n-1} C_{n-1}\right)-{ }^{n} C_{n}$
$={ }^{2 n} C_{n}-{ }^{n} C_{n}$
$\therefore W={ }^{2 n} C_{n}-1$