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$The sum of the digits of the number of positive integral solutions satisfying the equation (x_{1} + x_{2} + x_{3}) (y_{1} + y_{2}) = 77$

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Solution

$(x_{1} + x_{2} + x_{3}) (y_{1} + y_{2}) = 11 \times 7 or 7 \times 11$
$In the first case, (x_{1} + x_{2} + x_{3}) = 11 and (y_{1} + y_{2}) = 7, which have$
$^{10} C_{2} \cdot^{6} C_{1} solutions$
$In the second case, (x_{1} + x_{2} + x_{3}) = 7 and (y_{1} + y_{2}) = 11, which$
${have}^{6} C_{2} \cdot^{10} C_{1} solutions$
$∴ Total number of solutions =^{10} C_{2} \cdot^{6} C_{1} +^{6} C_{2} \cdot^{10} C_{1}$
$= 270 + 150 = 420$

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Combinations

Remember concepts with our Masterclasses.

The sum of the digits of the number of positive integral solutions satisfying the equation x1+x2+x3y1+y2=77

x1+x2+x3y1+y2=11×7 or 7×11 In the first case, x1+x2+x3=11 and y1+y2=7, which have 10C2⋅6C1 solutions In the second case, x1+x2+x3=7 and y1+y2=11, which have 6C2⋅10C1 solutions ∴ Total number of solutions =10C2⋅6C1+6C2⋅10C1 =270+150=420

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$The sum of the digits of the number of positive integral solutions satisfying the equation (x_{1} + x_{2} + x_{3}) (y_{1} + y_{2}) = 77$