First slide

Combinations

Question

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

Moderate

A

$^{2 n} C_{n} - 1$
B

$^{2 n - 1} C_{n} - 1$
C

$^{2 n + 1} C_{n - 1}$
D

$^{2 n - 1} C_{n - 1} - 1$

Solution

In general, we know that
For the distribution equation
$x_{1} + x_{2} + x_{3} + \dots + x_{n} \leq n$
Let required ways = W
$\Rightarrow W = \{No . of ways of distributing 1 item\} + \{No . of ways of distributing 2 items\} + . . . . . . + \{No . of Ways of distributing n items\}$
$\begin{array}{l} =^{1 + n - 1} C_{n - 1} +^{2 + n - 1} C_{n - 1} + \dots +^{n + n - 1} C_{n - 1} \\ =^{n} C_{n - 1} +^{n + 1} C_{n - 1} + \dots +^{2 n - 1} C_{n - 1} \\ = (^{n} C_{n - 1} +^{n} C_{n}) +^{n + 1} C_{n - 1} + \dots +^{2 n - 1} C_{n - 1} -^{n} C_{n} \\ = \{(^{n + 1} C_{n} +^{n + 1} C_{n - 1}) + \dots +^{2 n - 1} C_{n - 1}\} -^{n} C_{n} \end{array}$
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$\begin{array}{l} = (^{2 n - 1} C_{n} +^{2 n - 1} C_{n - 1}) -^{n} C_{n} \\ =^{2 n} C_{n} -^{n} C_{n} \\ ∴ W =^{2 n} C_{n} - 1 \end{array}$

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