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

Introduction / Context:
This question checks understanding of combinations and the use of complementary counting. We must choose a group of four children from a mixed group of boys and girls, under the condition that there is at least one boy in the selected group.

Given Data / Assumptions:

• Total boys = 6.
• Total girls = 4.
• We select 4 children in total.
• The selection should contain at least one boy.

Concept / Approach:
The simplest way is to first count all possible groups of four children without any restriction, and then subtract the number of groups that violate the condition. The only disallowed case is a group with no boys at all, that is, a group containing only girls.

Step-by-Step Solution:
Step 1: Total number of children = 6 + 4 = 10. Step 2: Total ways to choose any 4 children from 10 = C(10, 4). Step 3: Evaluate C(10, 4) = 10 * 9 * 8 * 7 / (4 * 3 * 2 * 1) = 210. Step 4: Now count the groups with no boys (only girls). We have 4 girls and need to select 4 of them. Step 5: Number of all-girl groups = C(4, 4) = 1. Step 6: Subtract these invalid groups from the total. Valid groups with at least one boy = 210 - 1 = 209.

Verification / Alternative check:
You could also count directly by cases (for example, 1 boy and 3 girls, 2 boys and 2 girls, 3 boys and 1 girl, and 4 boys and 0 girls) and add the results. Doing this carefully will also yield 209, which confirms the complementary counting method.

Why Other Options Are Wrong:
205, 194 and 159 arise if some valid distributions are omitted or if an incorrect total number of combinations is used. They do not match the value obtained either by complementary counting or by the casewise approach.

Common Pitfalls:
Many learners forget to subtract the all-girl case or accidentally subtract more cases than necessary. Others confuse combinations with permutations and incorrectly multiply by extra factors based on ordering, which is not relevant here since the order of children in the group does not matter.

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