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 picnic once, the total number of dolls that the girls have got is A. 21B. 45C. 27D. 24

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Hint: Here it is said that from the class of seven students five students go on a picnic. So, to find the students here we will use the concept of combinations to make different groups of students.

Complete step-by-step solution-

Now, according to the question, a group of five students go on a picnic every Saturday. Now, as there are 4 boys so there is at least one girl who is going on a picnic. Now, every Saturday a different group is sent so we have to find all the different groups. Also, to every girl on a picnic a doll is given so, by calculating the groups we can find the total number of dolls. Now the groups contain students as follows.
1). 4 boys and 1 girl
2). 3 boys and 2 girls
3). 2 boys and 3 girls
No other combination is possible according to the question. Now we have to find the total number of combinations with each group. We will use the formula ${}^n{C_r}$ = $\dfrac{{n!}}{{r!(n - r)!}}$ to find the number of combinations.

Now for group (1), all 4 boys are selected and 1 from 3 girls is to be selected. So,
Number of selections = ${}^4{C_4} \times {}^3{C_1}$ = 1 x 3 = 3 combinations.
So, for group (1), there are 3 combinations. Also, there is one girl in group (1) so, number of dolls in group (1) = 3 x 1 = 3 dolls.

Now, for group (2), 3 boys are to be selected from 4 boys and 2 girls from 3 girls are to be selected. So, the number of selections = ${}^4{C_3} \times {}^3{C_2}$ = 4 x 3 = 12 combinations.
So, according to the question , the number of dolls in group (2) = 12 x 2 = 24 dolls.

For group (3), 2 boys are to be selected from 4 boys and all 3 girls are selected. So,
Number of selections = ${}^4{C_2} \times {}^3{C_3}$ = 6 x 1 = 6 combinations.
So, according to the question number of dolls in group (3) = 6 x 3 = 18 dolls.
So, the total number of dolls that girls got = 3 + 24 + 18 = 45 dolls.

So, option (B) is correct.

Note: To solve such problems it is necessary that you make proper combinations. The main mistake students do in such types of problems is that they use the formula of permutation instead of using the formula of combination for finding total number of selections. Read the question properly to avoid such mistakes.