First slide

Combinations

Question

A class consists of 4 boys and g girls. Every Sunday five students, including at least three boys go for a picnic to Appu Ghar, a different group being 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 value of g is

Moderate

A

15
B

12
C

8
D

5

Solution

Number of groups having 4 boys and 1 girl
$= (^{4} C_{4}) (^{8} C_{1}) = g$
and number of groups having 3 boys and 2 girls
$= (^{4} C_{3}) (^{g} C_{2}) = 2 g (g - 1)$
Thus, the number of dolls distributed
$= g (1) + (2) [2 g (g - 1)] = 4 g^{2} - 3 g$
We are given $4 g^{2} - 3 g = 85 \Rightarrow g = 5$

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