Understanding the Problem of Athletic Teams

The question asks us to find the total number of members across three athletic teams: cricket, hockey, and football. We are given the number of members in each team individually, in pairs of teams, and in all three teams. This kind of problem is best solved using the principles of set theory, specifically the inclusion-exclusion principle for three sets.

Applying Set Theory to Athletic Teams

Let's represent the teams as sets:

• C = Set of members in the cricket team
• H = Set of members in the hockey team
• F = Set of members in the football team

We are given the following information:

• Number of members in the cricket team, $|C| = 21$
• Number of members in the hockey team, $|H| = 26$
• Number of members in the football team, $|F| = 29$
• Number of members who play hockey and cricket, $|H \cap C| = 14$
• Number of members who play hockey and football, $|H \cap F| = 15$
• Number of members who play football and cricket, $|F \cap C| = 12$
• Number of members who play all three games (cricket, hockey, and football), $|C \cap H \cap F| = 8$

We want to find the total number of members in the three athletic teams, which is the total number of members in the union of these three sets, $|C \cup H \cup F|$.

Using the Inclusion-Exclusion Principle

The principle of inclusion-exclusion for three sets is a fundamental concept in set theory used to find the number of elements in the union of three sets. The formula is:

$$ |C \cup H \cup F| = |C| + |H| + |F| - (|C \cap H| + |C \cap F| + |H \cap F|) + |C \cap H \cap F| $$

This formula accounts for members who are counted multiple times when we just sum the individual team memberships. We subtract the overlaps between pairs of sets, but this subtraction removes the members in all three sets too many times, so we add back the number of members in the intersection of all three sets.

Calculating the Total Members

Now, let's substitute the given values into the inclusion-exclusion formula to find the total number of members in the athletic teams:

$$ |C \cup H \cup F| = 21 + 26 + 29 - (14 + 12 + 15) + 8 $$

First, sum the individual team memberships:

$$ 21 + 26 + 29 = 76 $$

Next, sum the overlaps between pairs of teams:

$$ 14 + 12 + 15 = 41 $$

Now, plug these sums back into the formula:

$$ |C \cup H \cup F| = 76 - 41 + 8 $$

Perform the final calculations:

$$ 76 - 41 = 35 $$

$$ 35 + 8 = 43 $$

So, the total number of members in the three athletic teams is 43.

Using the inclusion-exclusion principle helps us accurately count each unique member exactly once, ensuring we find the correct total number of members across the different athletic teams.

Final Answer for Athletic Teams Membership

The total number of members in the three athletic teams (cricket, hockey, and football) is 43. This calculation demonstrates the practical application of set theory in solving problems involving overlapping groups.