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Range of Quadratic Expression

The number of...

Question

The number of non negative integral solutions to the system of equations $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} = 25$ and $a_{1} + a_{2} + a_{3} = 10$ are

2^{8} + 2^{5}

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2^{10} + 2^{5}

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2^{25} + 2^{10}

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32

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Solution

The correct option is B $2^{10} + 2^{5}$
PROBLEMS BASED ON CERTAIN THEOREMS ON COMBINATIONS :
$\cdot$ The number of positive integral solutions of the equation $x_{1} + x_{2} + x_{3} + . . . . + x_{r} = n$ is $^{n - 1} C_{r - 1} .$ $\cdot$ The number of non-negative integral solutions of the equation<br> $x_{1} + x_{2} + x_{3} + . . . . + x_{r} = n$ is $^{n + r - 1} C_{r - 1} .$

Consider the equations
$a_{1} + a_{2} + a_{3} + a_{4} + a_{5} = 25$ (1) and
$a_{1} + a_{2} + a_{3} = 10$ (2)
Hence we have, $a_{4} + a_{5} = 15$ (3)
The number of solutions for the system is same as number of solutions for the equation (2) times the number of solutions for the equation (3).
The number of solutions for equation (2) are $^{10 + 3 - 1} C_{3 - 1} =^{12} C_{2} = \frac{12!}{2! 10!} = 66$ .
The number of solutions of equation (3) are $^{15 + 2 - 1} C_{2 - 1} =^{16} C_{1} = \frac{16!}{1! 15!} = 16$ .
Hence the total number of solutions for the system are $66 \times 16 = (64 + 2) 16 = (2^{6} + 2) 2^{4} = 2^{10} + 2^{5}$ .

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Similar questions

a_{1} + a_{2} + a_{3} + a_{4} + a_{5} = 25

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