Question

A class contains $4$boys and $g$girls.

Every Sunday $5$students including at least $3$boys go for a picnic at the dollhouse, a different group is sent every week.

During the picnic, the class teacher gives each girl in the group a doll.

If the total number of dolls distributed was$85$, then the value of $g$is?

• A. $15$
• B. $12$
• C. $8$
• D. $5$

Answer 1

The correct option is D

$5$

Find the value of g:

Since we have to choose $5$ students out of which at least $3$ are boys so

we can choose $3$boys and $2$girls or $4$boys and $1$girl. And each girl is given $1$doll, Since the total number of dolls distributed was $85$

$\Rightarrow 2\times {}^{4}C_3\times {}^{g}C_2+\hspace{0.17em}1\times {}^{4}C_4\times {}^{g}C_1=85$ (since in $1st$ way there are $2$ girls and in $2nd$ way, there is only $1$girl)

$\Rightarrow 2\times \frac{4!}{3!}\times \frac{g(g-1)}{2}+1\times 1\times g=85\mathbf{}\mathbf{}\left[\because {}^{n}C_n=1\right]$

$$\begin{gathered} \Rightarrow 8\times \frac{g(g-1)}{2}+\hspace{0.17em}\hspace{0.17em}g=85 \\ \Rightarrow 4g(g-1)+\hspace{0.17em}g=85 \\ \Rightarrow 4g^2-3g-\hspace{0.17em}85=0 \\ \Rightarrow (g-5)(4g+17)=0 \\ \Rightarrow g=5 \end{gathered}$$

Hence the correct option is (D).