solve this using Polya's strategy without using formula, in a basketball league consisting of 10 teams, each team plays each of the other teams exactly three times. how many league games will be played.



Polya's strategy for problem-solving consists of four steps: (1) Understanding the problem, (2) Devising a plan, (3) Carrying out the plan, and (4) Looking back (checking the result). Let's apply this step-by-step approach to the given problem.

### 1. **Understanding the Problem:**
- We have 10 teams in a basketball league.
- Each team plays every other team exactly 3 times.
- We need to find how many league games will be played in total.

A key observation here is that a game occurs between two teams, and for every two teams, there are exactly 3 games played.

### 2. **Devising a Plan:**
- Consider that each pair of teams plays against each other three times.
- The number of ways to choose two teams from a set of 10 teams is a fundamental problem of combinations (which normally uses the formula \( \binom{n}{2} \), but we will not directly use the formula here).
- Once we know how many pairs of teams there are, we will multiply this by 3, since each pair plays 3 games.

### 3. **Carrying Out the Plan:**
Instead of using the formula, we will reason it out step by step:

- Start with one team (say, Team A). Team A will play against the other 9 teams.
- Team B will have already played against Team A, so Team B will play against the remaining 8 teams.
- Team C has already played against Teams A and B, so Team C will play against 7 remaining teams.
- This continues until the second-to-last team plays only against the last team.

So, the number of unique matchups is the sum of:
\[
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45
\]

Now, since each pair of teams plays 3 games, we multiply the number of unique matchups by 3:
\[
45 \times 3 = 135
\]

### 4. **Looking Back:**
We’ve reasoned through the problem without using formulas, confirming that the total number of games played is 135.

### Answer:
The total number of league games that will be played is **135**.

---

Would you like further details or explanations?

### Follow-up Questions:
1. How would the total number of games change if the teams played 4 times instead of 3?
2. What if there were 12 teams instead of 10, but still playing 3 games each?
3. Can you think of another method to solve this problem without using a formula?
4. How many games would be played if there were 8 teams, each playing 2 games against each other?
5. What would be the result if each team only played once with the others?

### Tip:
When counting pairwise interactions like this, it helps to think of it as building a "complete graph" where every node is connected to every other node.

In a basketball league consisting of 10 teams, each team plays each of the other teams exactly three times. How many league games will be played?

Learn how to calculate the total number of games in a basketball league with 10 teams, where each team plays every other team 3 times. Using combinatorics and reasoning without a formula, find out how 135 total league games are played. Suitable for Grades 9-12 students.

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