Problem 8 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Out of 9 outstanding students in a college, there are 4 boys and 5 girls. A team of four students is to be selected for a quiz programme. Find the probability that 2 are boys and 2 are girls. [CBSE-94]

Step-by-Step Solution

Answer

The probability is \( \frac{10}{21} \).

Step 1: Determine Total Ways to Select 4 Students

First, we calculate the total number of ways to select a team of 4 students from the 9 outstanding students. This can be done using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of students, and \( r \) is the number of students to be selected. Thus, we have:\[ \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

Step 2: Calculate Ways to Select 2 Boys

Next, we find the number of ways to select 2 boys from the 4 boys available. Again, we use the combination formula:\[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \]

Step 3: Calculate Ways to Select 2 Girls

Now, we find the number of ways to select 2 girls from the 5 girls available using the combination formula:\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

Step 4: Calculate Ways to Select a Team with 2 Boys and 2 Girls

We find the total number of ways to form a team of 4 students with 2 boys and 2 girls by multiplying the number of combinations from the previous steps:\[ 6 \times 10 = 60 \]

Step 5: Determine Probability of Selecting a Team with 2 Boys and 2 Girls

Finally, we determine the probability that the team of 4 students consists of 2 boys and 2 girls. This is the number of favorable outcomes divided by the total number of outcomes:\[ \text{Probability} = \frac{60}{126} = \frac{10}{21} \]

Key Concepts

Combinatorics

Combinatorics is a fascinating area of mathematics dealing with counting, arrangements, and combinations of objects. It's about understanding how we can select and arrange items to form certain structures. In this scenario, combinatorics helps us determine how many different teams of students we can form from a group. When we talk about arrangements without regard to order, we're referring to combinations. The beauty of combinatorics is in how it simplifies complex counting problems into more manageable calculations. By knowing the total number of items and those we wish to choose, we can easily find the number of possible combinations. This foundation is critical in solving problems relating to team selection, like the one in this exercise.

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics, widely used in counting combinations. It is denoted as \( \binom{n}{r} \), pronounced as 'n choose r', and represents the number of ways to choose \( r \) items from \( n \) items without considering the order of selection. It can be calculated using the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \(!\) denotes factorial, which is the product of all positive integers up to a given number. Thus, the binomial coefficient answers the fundamental question of how many ways we can pick a subset of items from a larger set. In this problem, it helps us compute the number of possible ways to select students to form a team, ensuring each step is logical and grounded in clear mathematical reasoning.

Probability Calculation

Probability refers to the measure of the likelihood that a particular event will occur. It is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Therefore, probability gives us insights into how likely it is for a certain arrangement, or selection, to happen. In mathematical terms, the probability \( P(E) \) of an event \( E \) occurring is given by:

• \( P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

In this task, we identify the probability of forming a desired combination of a team with exactly 2 boys and 2 girls. By identifying the number of favorable ways to choose such a team and comparing it to the overall number of team selections possible, we derive the probability using a simple yet powerful calculation.

Team Selection Problem

The team selection problem is a common scenario in combinatorics where we need to choose a specific number of members from a larger group based on certain criteria or constraints. In this exercise, we must pick 4 students with the condition that 2 should be boys and 2 should be girls. Solving this problem involves understanding and applying the concepts of combinations and probability effectively. By breaking down the task into smaller parts – first finding the number of combinations of boys and girls separately and then combining them – we simplify the whole process. It's like putting together pieces of a puzzle, where each step logically leads to a complete solution, ensuring we accurately solve the team selection problem.