Question

Number of positive integral solutions satisfying the equation $(x_1+x_2+x_3)(y_1+y_2)=77$, is

• A. 150
• B. 270
• C. 420
• D. 1024

Answer 1

The correct option is C 420

We have,

$(x_1+x_2+x_3)(y_1+y_2)=77$

$77=1\times 77=11\times 7$

As e need positive integral solutions

So,

$x_1+x_2+x_3=11$ and $y_1+y_2=7$

Or

$x_1+x_2+x_3=7$ and $y_1+y_2=11$

Number of positive integral solution of

$x_1+x_2+.......+x_n=k{.}^{k-1}{C}_{n-1}$

So, total number of solutions in this case.

${=}^{11-1}{C}_{3-1}{\times }^{7-1}{C}_{2-1}{+}^{7-1}{C}_{3-1}{\times }^{11-1}{C}_{2-1}$

${=}^{10}{C}_2{\times }^6{C}_1{+}^6{C}_2{\times }^{10}{C}_1$

$=270+150$

$=420$

Hence, this is the answer.