In a class there are 15 boys and 10 girls. Three students are selected at random. What is the probability that the selected group consists of exactly 1 girl and 2 boys?

Introduction / Context:
This is a selection problem involving two categories, boys and girls. The class has 15 boys and 10 girls, and three students are chosen without replacement. We are asked to find the probability that the group contains exactly 1 girl and 2 boys. This tests competence with combinations and probability based on equally likely subsets.

Given Data / Assumptions:

• Number of boys = 15.
• Number of girls = 10.
• Total students = 25.
• Three students are selected at random without replacement.
• We want exactly 1 girl and 2 boys.

Concept / Approach:
We use combinations to count the number of ways to choose any 3 students from 25. Then we count the number of favourable selections that contain exactly 1 girl and 2 boys, by choosing 1 girl from the 10 available and 2 boys from the 15 available. The probability is the ratio of favourable selections to total selections.

Step-by-Step Solution:
Total students = 25. Total number of ways to choose 3 students = C(25,3). Compute C(25,3) = 25 * 24 * 23 / (3 * 2 * 1) = 2300. Number of ways to choose exactly 1 girl from 10 = C(10,1) = 10. Number of ways to choose 2 boys from 15 = C(15,2). Compute C(15,2) = 15 * 14 / 2 = 105. Total favourable selections = 10 * 105 = 1050. Required probability = favourable / total = 1050 / 2300. Simplify 1050 / 2300 by dividing numerator and denominator by 50 to get 21 / 46.

Verification / Alternative check:
We can approximate the value 21/46 numerically as about 0.4565. This is slightly less than one half, which is reasonable given that there are more boys than girls, but the combination requirement restricts the composition. The numerator and denominator have no common factor other than 1, so 21/46 is in simplest form, confirming the correctness of simplification.

Why Other Options Are Wrong:
1/5: This is 0.2, which is too small given that having exactly 1 girl and 2 boys is a fairly likely outcome in this class.
3/25: This is 0.12 and does not correspond to the ratio 1050/2300.
1/50: This is 0.02, which is unrealistically small for this configuration.
None of these: This is incorrect because 21/46 matches the correct calculation and is listed among the options.

Common Pitfalls:
Common mistakes include counting permutations instead of combinations, or misreading the requirement and including cases with 0 or 2 girls. Another error is miscomputing C(25,3) or C(15,2) due to arithmetic slips. Systematic use of the combination formula and good arithmetic practices help to avoid these issues.

Final Answer:
Therefore, the probability that exactly 1 girl and 2 boys are selected is 21/46.