In a group of 6 boys and 4 girls, four children are to be selected. In how…

Concept: When a selection must satisfy an “at least one” condition, use complementary counting — find the total number of unrestricted selections and subtract the selections that violate the condition. This works because every unrestricted selection falls into exactly one of the two groups: those meeting the condition, and those that don't.

Application — this question:

1. Total children available: 6 boys + 4 girls = 10. Choosing any 4 of them without restriction gives 10C4 ways.

2. 10C4 = 10! / (4! × 6!) = 210.

3. The “at least one boy” condition is violated only when all 4 selected children are girls — that means choosing all 4 available girls: 4C4 = 1 way.

4. Ways with at least one boy = Total − All-girls case = 210 − 1 = 209.

Cross-check — case-by-case (boys count vs. girls count) confirms the same total:

Sum of all cases = 24 + 90 + 80 + 15 = 209, matching the complementary-counting result above.