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The problem you're asking about deals with a scenario where 20 football games are played, and for each game, there are three possible outcomes: win, lose, or tie. The question asks how many ways a football prognosticator can predict exactly 15 games correctly out of the 20.

Here's how we can approach this:

### Step-by-Step Solution:

1. **Correct Predictions**:
   The prognosticator predicts the outcome of all 20 games. They correctly predict exactly 15 of them. This means they incorrectly predict the other 5 games.

2. **Number of Ways to Choose the Correct Games**:
   The number of ways to choose 15 correct games out of 20 is given by a combination formula:
   \[
   \text{Number of ways to choose 15 games from 20} = \binom{20}{15} = \binom{20}{5}
   \]
   The reason \(\binom{20}{15} = \binom{20}{5}\) is that choosing 15 correct games out of 20 is equivalent to choosing 5 incorrect games (the complement of the correct ones).

3. **Predictions for Incorrect Games**:
   For each of the 5 incorrect games, there are 2 possibilities for the prognosticator (since one of the 3 possible outcomes must be wrong, and that leaves 2 incorrect choices for each game).

   Therefore, the number of ways to choose incorrect predictions for those 5 games is:
   \[
   2^5
   \]

4. **Total Number of Ways**:
   The total number of ways to achieve exactly 15 correct predictions is the product of the number of ways to choose the 15 correct games and the number of ways to predict incorrectly for the other 5 games. This gives us:
   \[
   \text{Total} = \binom{20}{15} \times 2^5 = \binom{20}{5} \times 2^5
   \]

Let me calculate this.The total number of ways the prognosticator can correctly predict the outcome of exactly 15 games out of 20 is **496,128**.

Would you like further details or explanations on any part of the solution?

Here are five related questions you might find interesting:
1. What would the total be if the prognosticator had to predict exactly 10 correct games out of 20?
2. How would the answer change if each game had 4 possible outcomes instead of 3?
3. Can you calculate the total number of possible prediction outcomes if no restrictions were placed on the number of correct guesses?
4. How does the combination formula relate to Pascal's Triangle?
5. What if the prognosticator only got 1 game wrong, how many ways could that happen?

**Tip:** The binomial coefficient \(\binom{n}{k}\) is symmetric, meaning \(\binom{n}{k} = \binom{n}{n-k}\), which simplifies certain calculations!

On a certain weekend, 20 football games are played, and for each game, three outcomes, win, lose, or tie, are possible. In how many ways can a football prognosticator predict one of these three outcomes for each game and end up predicting correctly on exactly 15 games?

Explore how to calculate the number of ways a football prognosticator can predict the outcome of 20 football games and get exactly 15 predictions correct. This problem involves combinatorics, using the binomial coefficient and probability theory. Suitable for students in Grades 10-12.

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