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 question involves combinations and mixed groups from two categories, boys and girls. It asks for the probability that a randomly chosen group of three students has exactly one girl and two boys, so we must count favourable and total combinations carefully and then form the probability fraction.

Given Data / Assumptions:

• Boys = 15.
• Girls = 10.
• Total students = 25.
• We select 3 students at random without replacement.
• Event of interest: exactly 1 girl and 2 boys.

Concept / Approach:
Order does not matter in forming a group, so we use combinations. The total number of possible groups of 3 from 25 students is C(25, 3). The favourable groups must have 1 girl chosen from 10 and 2 boys chosen from 15. The probability is favourable combinations divided by total combinations.

Step-by-Step Solution:
Total groups of 3 from 25 = C(25, 3). C(25, 3) = 25 * 24 * 23 / (3 * 2 * 1) = 2300. Ways to choose 1 girl from 10 = C(10, 1) = 10. Ways to choose 2 boys from 15 = C(15, 2). C(15, 2) = 15 * 14 / 2 = 105. Favourable groups = 10 * 105 = 1050. Probability = 1050 / 2300. Simplify by dividing numerator and denominator by 50: 1050 / 2300 = 21 / 46.

Verification / Alternative check:
We can also use a stepwise probability approach. First pick a girl then two boys or in any order, but account for permutations. However, summing over all orders will return the same result once we simplify, which acts as a check on the combination method. The combination method is simpler and less error prone here.

Why Other Options Are Wrong:
1/5 and 3/25 are much smaller than the correct fraction and do not correspond to any natural counting in this context. 1/50 is far too small and would suggest very few favourable groups, which contradicts the large number of ways to choose 2 boys and 1 girl.

Common Pitfalls:
Learners sometimes confuse exactly 1 girl with at least 1 girl and accidentally include cases of 2 or 3 girls. Another mistake is to use permutations instead of combinations or to forget to multiply the choices for boys and girls. Keeping track of the structure "choose girls then boys" helps avoid these errors.

Final Answer:
Therefore, the probability of selecting exactly 1 girl and 2 boys is 21/46.