Question

A bag contains $5$ blue balls and unknown number of red balls, two balls are drawn at random. The probability of both of them are blue is $\frac{5}{14}$, then the number of red balls are

• A. $3$
• B. $2$
• C. $4$
• D. $5$

Answer 1

The correct option is A $3$

Let the number of red balls be x.

According to the given question,
$\frac{{}^5{C}_2}{{}^{5+x}{C}_2}=\frac{5}{14}$

$\Rightarrow \frac{\frac{20}{2}}{\frac{(5+x)!}{2(3+x)!}}=\frac{5}{14}$

$\Rightarrow \frac{20(3+x)!}{(5+x)!}=\frac{5}{14}$

$\Rightarrow 56=(5+x)(4+x)$

$\Rightarrow 56=x^2+20+9x$

$\Rightarrow x^2+9x-36=0$

$\Rightarrow x^2+12x-3x-36=0$

$\Rightarrow x=3$

$\therefore$ The number of red balls are $3$.