Two different families A and B are blessed wi | vedclass

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Question

(D) Let the number of children in each family be $x$.The total number of children in both families is $2x$.We are distributing $3$ tickets among $2x$

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(D) Let the number of children in each family be $x$.The total number of children in both families is $2x$.We are distributing $3$ tickets among $2x$ children such that no child gets more than one ticket.The total number of ways to choose $3$ children out of $2x$ is $^{2x}C_{3}$.The number of ways to choose $3$ children from family $B$ (which has $x$ children) is $^{x}C_{3}$.The probability that all $3$ tickets go to children of family $B$ is given by:$P = \frac{^{x}C_{3}}{^{2x}C_{3}} = \frac{1}{12}$Expanding the combinations:$\frac{\frac{x(x-1)(x-2)}{3!}}{\frac{2x(2x-1)(2x-2)}{3!}} = \frac{1}{12}$$\frac{x(x-1)(x-2)}{2x(2x-1)(2x-2)} = \frac{1}{12}$$\frac{(x-1)(x-2)}{2(2x-1) \cdot 2(x-1)} = \frac{1}{12}$$\frac{x-2}{4(2x-1)} = \frac{1}{12}$$\frac{x-2}{2x-1} = \frac{4}{12} = \frac{1}{3}$$3(x-2) = 2x-1$$3x - 6 = 2x - 1$$x = 5$Thus,the number of children in each family is $5$.

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