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If x_1+x_2+x_3+x_4+x_5=20 and x_1+x_2=5,\left(x_1, x_2, x_3, x_4, x_5 \geq\right. 0) then find the number of non negative integral solutions of above equation.

Option: 1

804


Option: 2

768


Option: 3

824


Option: 4

816


Answers (1)

best_answer

\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4+\mathrm{x}_5=20, \mathrm{x}_1+\mathrm{x}_2=5 \\

\Rightarrow \mathrm{x}_3+\mathrm{x}_4+\mathrm{x}_5=15

Number of solutions

\Rightarrow Coefficient of x^5 in (1) \times coefficient of x^{15} in (2)

\Rightarrow Coefficient of  x^5\left(\frac{1-x^6}{1-x}\right)^2 \times Coefficient of                                                       x^{15}\left(\frac{1-x^{16}}{1-x}\right)^3 \\

\Rightarrow Coefficient of x^5  in (1-x)^{-2} \times Coefficient of x^{15}(1-x)^{-3} \\

{ }^{2+5-1} C_1 \times{ }^{3+15-1} C_{3-1}={ }^6 C_1 \times{ }^{17} C_2

            =\frac{6 \times 17 \times 16}{2}=48 \times 17=816

Posted by

Ajit Kumar Dubey

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