Combinatorial Analysis of Selecting Students with at Least Two Girls

Introduction

When faced with the task of selecting a group of students from a class containing a specific gender distribution, combinatorial analysis becomes a useful tool. This article delves into the process of selecting 5 students from a class consisting of 4 boys and 6 girls such that at least 2 of the selected students are girls. We will explore different cases, calculate the number of valid combinations for each, and finally sum them up to find the total number of ways to achieve the desired outcome.

Methodology

We begin by breaking down the problem into different cases based on the number of girls selected. These cases are:

Selecting 2 girls and 3 boys Selecting 3 girls and 2 boys Selecting 4 girls and 1 boy Selecting 5 girls with no boys

We will use combinatorial formulas to calculate the number of ways to form each of these groups and then sum them up to get the total number of valid combinations.

Case 1: Selecting 2 Girls and 3 Boys

To select 2 girls from 6, we use the binomial coefficient (binom{6}{2}). For 3 boys from 4, we use (binom{4}{3}). The total number of ways to form this group is:

(binom{6}{2} times binom{4}{3} 15 times 4 60)

Case 2: Selecting 3 Girls and 2 Boys

Similarly, to select 3 girls from 6, we use (binom{6}{3}), and for 2 boys from 4, we use (binom{4}{2}). Thus, the total number of ways is:

(binom{6}{3} times binom{4}{2} 20 times 6 120)

Case 3: Selecting 4 Girls and 1 Boy

To select 4 girls from 6, we use (binom{6}{4}), and for 1 boy from 4, we use (binom{4}{1}). Therefore, the total number of ways to form this group is:

(binom{6}{4} times binom{4}{1} 15 times 4 60)

Case 4: Selecting 5 Girls with No Boys

Finally, to select 5 girls from 6, we use (binom{6}{5}). This gives us:

(binom{6}{5} 6)

Total Combinations

Summing up all the valid combinations from each case, we get:

(60 120 60 6 246)

This means there are 246 ways to select 5 students from a class of 4 boys and 6 girls with at least 2 girls included.

Alternative Approach: Probability Calculation

Alternatively, the probability can be calculated to determine the likelihood of selecting at least 2 girls in a random selection of 5 students. We first calculate the number of ways to select 2 girls and 3 boys, which is 60, and then divide this by the total number of ways to select any 5 students from 10 (6 girls 4 boys), which is (binom{10}{5} 252).

The probability is thus:

(frac{60}{252} frac{10}{42} frac{5}{21} approx 0.2381)

This means that approximately 23.81% of the time, at least 2 girls will be selected in a random group of 5 students.

Conclusion

Through combinatorial analysis and probability, we have determined the number of ways to select 5 students from a class of 4 boys and 6 girls with at least 2 girls included. This method can be applied to similar problems in various fields, such as statistics, engineering, and computer science, where understanding the likelihood of specific outcomes is crucial.