Question

A class contains three girls and four boys. Every Saturday, five go on a picnic (a different group of students is sent every week). During the picnic, each girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for a picnic once, the total number of dolls that the girls have got is?
A) \[21\]
B) \[45\]
C) \[27\]
D) \[24\]

Answer 1

Hint: To solve this question, i.e., to find the total number of dolls that the girls have got. We need to take three cases. We will consider the first case as when only one girl is present in the group, second case when two girls are present in the group and third case when three girls are present in the group, then on applying permutation n combination formula, we get our required answer.

Complete step-by-step answer:
We have been given that a class contains three girls and four boys and every Saturday, five go on a picnic (a different group of students is sent every week). It is given that during the picnic, each girl in the group is given a doll by the accompanying teacher. So, if all possible groups of five have gone for a picnic once, we need to find the total number of dolls that the girls have got.
So, a group may consist of \[1\] girl, \[2\] girls or \[3\] girls. That means there are three possible cases.
To find that, we will apply the combination formula, which is, \[^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}\]
\[1.\] When there is only one girl is present in the group, number of groups
\[
  { = ^3}{C_1}{ \times ^4}{C_4} \\
   = \dfrac{{3!}}{{1! \times (2)!}} \times \dfrac{{4!}}{{4! \times 0!}} \\
   = {\text{ }}3 \\
\]
\[2.\] When there are two girls are present in the group, number of groups
\[
  { = ^3}{C_2}{ \times ^4}{C_3} \\
   = \dfrac{{3!}}{{2! \times 1!}} \times \dfrac{{4!}}{{3! \times 1!}} \\
   = {\text{ }}3 \times 4 \\
   = 12 \\
\]

\[3.\] When there are three girls are present in the group, number of ways
\[
   = 1{ \times ^4}{C_2} \\
   = 1 \times \dfrac{{4!}}{{2! \times 2!}} \\
   = {\text{ 1}} \times 6 \\
   = 6 \\
\]
Therefore, total number of dolls \[ = 1 \times 3 + 2 \times 12 + 3 \times 6 = 45\;\;\]
So, the correct answer is “Option B”.

Note: Here, in the solution we have applied the combination formula i.e., \[^n{C_r},\] because combination is the number of ways of selecting the items in which the orders do not matter. So, in the solution we have considered three cases, and in individual cases, we made a combination of girls to be present in the group.