Problem 6 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In a group of students, there are 3 boys and 3 girls. Four students are to be selected at random from the group. Find the probability that either 3 boys and 1 girl or 3 girls and 1 boy are selected.

Step-by-Step Solution

Answer

The probability is \( \frac{2}{5} \).

Step 1: Determine Total Number of Ways to Select 4 Students

The first step is to determine how many ways we can choose 4 students from the total 6 students (3 boys and 3 girls). This can be calculated using combinations. The formula for combinations is \( C(n, k) = \frac{n!}{k!(n-k)!} \). Here, \( n = 6 \) and \( k = 4 \), so \( C(6, 4) = \frac{6!}{4!2!} = 15 \).

Step 2: Calculate Ways to Select 3 Boys and 1 Girl

We need to calculate the number of ways to select 3 boys from 3 and 1 girl from 3. The number of ways to select 3 boys is \( C(3, 3) = 1 \) and the number of ways to select 1 girl from 3 is \( C(3, 1) = 3 \). Therefore, the total ways to select 3 boys and 1 girl is \( 1 \times 3 = 3 \).

Step 3: Calculate Ways to Select 3 Girls and 1 Boy

Similarly, calculate the number of ways to select 3 girls from 3 and 1 boy from 3. The number of ways to select 3 girls is \( C(3, 3) = 1 \) and the number of ways to select 1 boy from 3 is \( C(3, 1) = 3 \). Therefore, the total ways to select 3 girls and 1 boy is \( 1 \times 3 = 3 \).

Step 4: Add Probabilities for Each Scenario

The total number of favorable outcomes, either 3 boys and 1 girl or 3 girls and 1 boy, is \( 3 + 3 = 6 \). The probability of either event occurring is the number of favorable outcomes divided by the total number of ways to select 4 students. Hence, the probability is \( \frac{6}{15} \).

Step 5: Simplify the Probability Fraction

Simplify the fraction \( \frac{6}{15} \). Divide the numerator and the denominator by their greatest common divisor, which is 3, to get \( \frac{2}{5} \).

Key Concepts

Combinatorics

Combinatorics is a fascinating field of mathematics that revolves around counting and arranging objects. It's like a mathematical toolbox filled with tools to solve problems regarding how many different ways things can happen. You'll often encounter combinatorics when dealing with problems where selection and arrangement matter, like our task here with students in a group.

In its simplest form, combinatorics helps you figure out things like:

• How many different ways can you roll a set of dice?
• How many ways can you line up people in a row?
• How many ways can you choose a committee from a larger group?

For our scenario with the group of students, combinatorics is used to calculate how many ways we can select a particular number of students from the group. Understanding the basics of combinatorics sets the foundation for tackling more complex problems, where you consider not only how many ways an event can occur, but also the likelihood or probability of such events.

Combinations and Permutations

When diving into the fascinating world of combinatorics, two key concepts you'll come across often are combinations and permutations. They are like two sides of a coin, each telling us about different ways to organize or select items.

• Permutations: These concern sequences or arrangements of items, where the order does matter. Think of it as arranging books on a shelf; different orders give different permutations.
• Combinations: When the order doesn't matter, as in selecting members for a team, we're talking about combinations. Here, it's all about the combination of items, regardless of which comes first or last.

For example, in our exercise, we use combinations because we're simply choosing a group of students, where the order in which we pick them doesn't change the outcome. The formula for combinations, \( C(n, k) = \frac{n!}{k!(n-k)!} \), helps us calculate how many ways we can select a certain number of items from a larger set, precisely what we need here for selecting students from the total group.

Binomial Coefficient

The binomial coefficient is a fundamental part of combinatorics used widely when calculating combinations. It's often represented using the notation \( \binom{n}{k} \), which is just another way to express the combination formula \( C(n, k) \).

The binomial coefficient tells us the number of ways to choose \( k \) items from \( n \) without caring about the order, echoing the idea behind combinations. It is also called "n choose k" because it represents choosing \( k \) items from a larger set of \( n \) total items. For example, when we calculate \( C(6, 4) \) to find how many ways we can pick 4 students out of 6, the result is given by the binomial coefficient. This is pivotal in probability theory, as it allows us to solve problems like our student selection by providing the number of favorable selections.

• Empowers you to determine probabilities in complex situations by providing a clear method to count selections.
• Extremely useful in statistics and various fields where hypotheses or outcomes need to be explored in terms of probability.