In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can the selection be made so that at least one boy is included among the four selected children?

Introduction / Context:
This is a classic selection problem involving a group of boys and girls with a condition that prohibits selecting all girls. It focuses on using combinations and the complement method, which is often simpler than direct counting of all allowed cases. Such questions are very common in the combinations part of aptitude tests.

Given Data / Assumptions:

• There are 6 boys.
• There are 4 girls.
• Total children available: 10.
• We need to select 4 children.
• The selection must contain at least one boy, so an all girl selection is not allowed.

Concept / Approach:
Instead of counting all valid combinations directly by splitting into cases based on how many boys are chosen, it is simpler to count all possible selections of 4 children from the 10 and then subtract the number of invalid selections, i.e., those that contain no boys (all girls). This is a straightforward application of the complement principle in counting.

Step-by-Step Solution:
Step 1: Total ways to choose any 4 children from 10 is 10C4.Step 2: Compute 10C4 = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 210.Step 3: To violate the condition, a selection would have to consist of girls only, that is 4 girls with no boys.Step 4: The number of ways to choose 4 girls from the 4 available girls is 4C4 = 1.Step 5: Therefore, the number of valid selections with at least one boy is 210 - 1 = 209.Step 6: Hence, there are 209 different ways to select 4 children with at least one boy.
Verification / Alternative check:
We can verify this briefly by counting in cases: exactly 1 boy, 2 boys, 3 boys, and 4 boys. The total would be 6C1 × 4C3 + 6C2 × 4C2 + 6C3 × 4C1 + 6C4 × 4C0. Computing these gives 6 × 4 + 15 × 6 + 20 × 4 + 15 × 1 = 24 + 90 + 80 + 15 = 209. This matches our complement method result and confirms correctness.

Why Other Options Are Wrong:
210 is the total number of ways without the restriction and thus includes the one invalid all girl selection. 200, 208 and 290 do not match either the complement calculation or the detailed case based counting and therefore are incorrect. Only 209 satisfies the constraint and matches both methods of calculation.

Common Pitfalls:
Students sometimes forget to exclude the all girl case or miscalculate combinations, especially 10C4. Another common mistake is to misinterpret the phrase "at least one boy" and treat it as "exactly one boy," which yields a much smaller number. Always read the condition carefully and choose the simplest counting strategy, which here is the complement method.

Final Answer:
The number of ways to select 4 children with at least one boy is 209.