the results of 11 chess matches (as win , lose or draw ) are to be forecast . Out of all posible forecasts, the number of ways in which 8 correct and 3 incorrect results can be forecast is

Options

A. `""^(11)C_(8)xx8`
B. `2^(3)xx""^(11)C_(3)`
C. 1320
D. 8! `xx` 3!

Answer

To solve the problem of forecasting the results of 11 chess matches, where we want to find the number of ways to forecast 8 correct and 3 incorrect results, we can break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have 11 matches.
- Each match can end in one of three outcomes: win, lose, or draw.
- We need to forecast 8 matches correctly and 3 incorrectly.

2. **Calculating Correct Forecasts**:
- For each match that is forecasted correctly, there is only 1 way to predict it (since it must match the actual outcome).
- Therefore, the number of ways to forecast 8 matches correctly is:
\[ 1^8 = 1 \]

3. **Calculating Incorrect Forecasts**:
- For each match that is forecasted incorrectly, there are 2 possible outcomes (since it can either be a win or a draw if you predicted a loss, and vice versa).
- Therefore, the number of ways to forecast 3 matches incorrectly is:
\[ 2^3 = 8 \]

4. **Choosing Which Matches to Forecast Correctly**:
- We need to choose 8 matches out of the 11 to be correct. The number of ways to choose 8 matches from 11 is given by the combination formula:
\[ \binom{11}{8} = \binom{11}{3} \]
- This is because choosing 8 correct matches is equivalent to choosing 3 incorrect matches.

5. **Combining the Results**:
- The total number of ways to forecast 8 correct and 3 incorrect results is given by multiplying the number of ways to choose the matches by the number of ways to forecast them correctly and incorrectly:
\[ \text{Total Ways} = \binom{11}{3} \times 2^3 \]

6. **Calculating the Values**:
- First, calculate \(\binom{11}{3}\):
\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \]
- Now, multiply this by \(2^3\):
\[ \text{Total Ways} = 165 \times 8 = 1320 \]

### Final Answer:
The number of ways in which 8 correct and 3 incorrect results can be forecasted is **1320**.

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