The result of 8 matches (win,loss or draw) are to be predicted. The nu...
Number of different forecasts containing exactly 6 correct results
There are 3 possible outcomes for each match - win, loss or draw. Therefore, there are 3^8 = 6561 possible forecasts for the 8 matches.
Calculating the number of forecasts with exactly 6 correct results
In order to calculate the number of forecasts with exactly 6 correct results, we need to consider the number of ways we can choose 6 out of the 8 matches to predict correctly. This can be calculated using the combination formula:
nCr = n! / (r! * (n-r)!)
where n is the total number of matches (8), and r is the number of matches we want to predict correctly (6).
Using this formula, we get:
8C6 = 8! / (6! * (8-6)!) = 28
Calculating the number of forecasts with 2 incorrect results
Now that we know there are 28 possible forecasts with exactly 6 correct results, we need to calculate the number of forecasts with 2 incorrect results. In order to do this, we need to consider the number of ways we can choose 2 out of the 8 matches to predict incorrectly. This can be calculated using the combination formula:
nCr = n! / (r! * (n-r)!)
where n is the total number of matches (8), and r is the number of matches we want to predict incorrectly (2).
Using this formula, we get:
8C2 = 8! / (2! * (8-2)!) = 28
Calculating the total number of forecasts with exactly 6 correct results
Now that we know there are 28 possible forecasts with exactly 6 correct results and 28 possible forecasts with 2 incorrect results, we can calculate the total number of forecasts with exactly 6 correct results:
Total number of forecasts with exactly 6 correct results = 28 * 28 = 784
Conclusion
Therefore, there are 784 different forecasts containing exactly 6 correct results.