From 4 children, 2 women, and 4 men (total 10 people), 4 persons are chosen uniformly at random. What is the probability that exactly 2 of the selected persons are children?

Introduction / Context:
This is a hypergeometric selection: we draw a fixed-size subset without replacement from distinct groups and ask for exactly k from one group.

Given Data / Assumptions:

• Total population N = 10 (4 children, 6 adults).
• Sample size n = 4.
• Event: exactly 2 children in the sample.

Concept / Approach:
Count ways to pick 2 from the 4 children and 2 from the 6 adults; divide by total ways to pick any 4 from 10.

Step-by-Step Solution:
Favorable = C(4, 2) * C(6, 2) = 6 * 15 = 90.Total = C(10, 4) = 210.Probability = 90/210 = 3/7 = 9/21.

Verification / Alternative check:
Reduce 90/210 by dividing numerator and denominator by 30 to obtain 3/7; the option list uses 21ths, so 9/21 matches.

Why Other Options Are Wrong:
Other fractions do not simplify to 3/7 given these counts.

Common Pitfalls:
Accidentally choosing 3 from one group and 1 from the other or forgetting to multiply the group combinations.

Final Answer:
9/21