The number of non-negative solutions of x1+x2+x3+, ldots,+xn ≤ n (where n is positive integer) is

Question

The number of non-negative solutions of x1+x2+x3+, ldots,+xn ≤ n (where n is positive integer) is (A)  2 n Cn-1 (B)  2 n-1 Cn-1 (C)  2 n+1 Cn-1 (D)  2 n-1 Cn-1-1

Tardigrade Admin · Accepted Answer

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 ⎭ ⎬ ⎫ + ⎩ ⎨ ⎧ c No. of ways of distributing 2 items ⎭ ⎬ ⎫ + \dots + ⎩ ⎨ ⎧ No. of Ways of distributing n items ⎭ ⎬ ⎫

=^{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}

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= (^{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

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

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

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

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

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

Q. The number of non-negative solutions of x1​+x2​+x3​+,…,+xn​≤n (where n is positive integer) is

2nCn​−1

2n−1Cn​−1

2n+1Cn−1​

2n−1Cn−1​−1

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

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

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

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

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