Problem 49 - Algebra for JEE ADVANCED

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. 21 b. 45 c. 27 d. 24

Step-by-Step Solution

Answer

b. 45

Step 1: Determine Total Possible Groups

We need to find the total number of ways to choose 5 students out of 7 (3 girls + 4 boys). This is a combination problem and can be solved using the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 7 \) and \( r = 5 \). Calculating gives \( \binom{7}{5} = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \). Thus, there are 21 different possible groups.

Step 2: Calculate Dolls per Group

Each group that goes on a picnic will include a certain number of girls. For each girl in the group, one doll is given. We need to determine how many dolls are distributed in total based on how many girls are in each group.

Step 3: Group Combinations with One Girl

Find groups that have 1 girl and 4 boys. Select 1 girl from 3, which is \( \binom{3}{1} = 3 \) ways, and select 4 boys from 4, which is \( \binom{4}{4} = 1 \) way, giving us \( 3 \times 1 = 3 \) groups. Each group contributes 1 doll.

Step 4: Group Combinations with Two Girls

Find groups that have 2 girls and 3 boys. Select 2 girls from 3, which is \( \binom{3}{2} = 3 \) ways, and select 3 boys from 4, which is \( \binom{4}{3} = 4 \) ways, giving us \( 3 \times 4 = 12 \) groups. Each group contributes 2 dolls.

Step 5: Group Combinations with Three Girls

Find groups that include all 3 girls and 2 boys. This is selecting 2 boys from the available 4, which is \( \binom{4}{2} = 6 \) ways. Each of these 6 groups contributes 3 dolls.

Step 6: Calculate Total Dolls

Now, calculate the total number of dolls distributed: \( 3 \times 1 = 3 \) dolls from one-girl groups, \( 12 \times 2 = 24 \) dolls from two-girl groups, and \( 6 \times 3 = 18 \) dolls from three-girl groups. Therefore, the total is \( 3 + 24 + 18 = 45 \) dolls.

Key Concepts

Combination formula

In combinatorics, the combination formula is used to determine the number of different ways you can choose a subset of "r" elements from a larger set of "n" items without regard to the order of the items. This is expressed with the formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \).

To understand how this works, consider the original exercise: a class with 3 girls and 4 boys, making a total of 7 students. We need to form groups of 5 to go on a picnic. By applying the combination formula, we set \( n = 7 \) and \( r = 5 \).

• The calculation becomes \( \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21 \).

This tells us there are 21 unique groups of 5 students that can be chosen from the class of 7.

Group selection

Group selection involves finding various subsets based on specific conditions. In our exercise, the problem revolves around selecting groups of 5 from a mix of boys and girls, and then counting certain combinations where a specific number of girls are included in each group.

To solve this, we identify the different group selection scenarios:

• Groups with 1 girl and 4 boys: Select 1 girl from 3, which gives \( \binom{3}{1} = 3 \) ways, and select the remaining 4 boys without choice, \( \binom{4}{4} = 1 \) way.
• Groups with 2 girls and 3 boys: Here, select 2 girls from 3, \( \binom{3}{2} = 3 \) ways, and 3 boys from 4, \( \binom{4}{3} = 4 \) ways.
• Groups with 3 girls and 2 boys: All girls are chosen, leaving the choice to 2 boys from 4, \( \binom{4}{2} = 6 \) ways.

Each scenario gives a different number of groups, which is crucial for determining further calculations, such as the total number of dolls distributed.

Problem-solving strategies

Problem-solving in combinatorics often involves breaking down a complex problem into smaller, manageable parts. Let's apply this method to the picnic problem by simplifying each step.

The key strategy is:

• First, calculate the total number of groups using the combination formula.
• Next, identify all possible group configurations by considering different numbers of girls in each group.
• Then, calculate the contributions to the final result (number of dolls) based on these configurations.

By systematizing the problem, step-by-step, you can ensure accuracy and completeness in your answer. This methodical approach illuminates the relationships between different elements in the problem, eliminating confusion and leading to a clear understanding.

Algebraic expressions

Algebraic expressions are used to represent the solutions in combinatorial problems. They express relationships between elements numerically, helping us compute outcomes efficiently.

In our exercise, the algebraic expressions derived from combinations include:

• \( 3 \times 1 = 3 \), representing the dolls when selecting 1 girl and 4 boys.
• \( 12 \times 2 = 24 \), for groups with 2 girls and 3 boys.
• \( 6 \times 3 = 18 \), for groups with 3 girls and 2 boys.

Algebra makes it easier to organize these calculations and derive the total sum of 45 dolls distributed. By using expressions, you simplify and reduce the risk of errors in your calculations, paving the way for more confident problem-solving in algebraic scenarios.