In a league there are 8 teams and each team must play every other team exactly three times during the season. How many matches will be played in total in the league?

Introduction / Context:
This question is a standard application of combinations in counting pairwise matches in a league. When teams in a round robin tournament all play one another, each match corresponds to an unordered pair of distinct teams. The twist here is that each pair of teams plays not once but three times. The goal is to determine the total number of matches played across the entire league.

Given Data / Assumptions:

• Total number of teams in the league = 8.
• Each pair of distinct teams plays exactly three matches against each other.
• There are no extra matches beyond these scheduled ones.
• Matches are counted without regard to home or away status; each individual game is one match between two teams.

Concept / Approach:
First we count how many different pairs of teams can be formed from 8 teams. This is a combinations problem because a match between Team A and Team B is the same as a match between Team B and Team A. Once we know the number of unique pairs, we multiply by 3 because each pair plays three matches. This is a straightforward use of the combination formula C(n, 2) for counting unordered pairs.

Step-by-Step Solution:
Step 1: Compute the number of distinct pairs of teams. Number of ways to choose 2 teams from 8 = C(8, 2). C(8, 2) = 8 * 7 / 2 = 28. Step 2: Each pair of teams plays exactly 3 matches. So, total matches = number of pairs * matches per pair. Total matches = 28 * 3 = 84.

Verification / Alternative check:
You can also view this as first thinking about a single round robin where each pair meets once, giving 28 matches. The question states that they play each other three times, which is equivalent to three such round robins. Thus, the total number of matches is 3 times 28, again confirming 84 as the correct count.

Why Other Options Are Wrong:
36 is too small and corresponds roughly to counting only some of the pairs or using the wrong formula. The number 72 would be correct if each pair played only about 2.57 matches, which is not meaningful here. The value 79 is arbitrary and does not arise from any simple combination calculation. Similarly, 96 would correspond to 32 pairs, but there are only 28 distinct pairs possible among 8 teams.

Common Pitfalls:
A typical mistake is to confuse ordered and unordered selections and incorrectly use permutations instead of combinations. Another common error is to forget to multiply by 3 after counting the pairs once. Some students may also try to count matches per team individually and double count, which leads to an incorrect total. Remembering that each match is associated with a unique unordered pair and then scaling by the number of games per pair avoids these problems.

Final Answer:
The league will have a total of 84 matches played.