To solve this problem, we need to consider the restriction that Ross (R) and Lionel (L) cannot be in the team together.
Method 1: Consider the cases separately:
- Choose R but not L: Choose 3 more people from the remaining 8, which is \\(^8C_3 = 56\\) ways.
- Choose L but not R: Choose 3 more people from the remaining 8, which is \\(^8C_3 = 56\\) ways.
- Choose neither R nor L: Choose all 4 people from the remaining 8, which is \\(^8C_4 = 70\\) ways.
Summing these scenarios gives: \\((56 + 56 + 70 = 182)\\) ways.
Method 2: Use the complement principle:
- Calculate the total number of ways to choose 4 people from 10 without restriction: \\(^10C_4 = 210\\) ways.
- Subtract the number of ways to choose both R and L in the team: Choose 2 more people from the remaining 8, which is \\(^8C_2 = 28\\) ways.
Thus, the number of valid ways is \\(210 - 28 = 182\\) ways.
Method 3: Consider excluding R or L:
- Exclude R: Choose 4 people from the remaining 9, which is \\(^9C_4 = 126\\) ways.
- Exclude L: Choose 4 people from the remaining 9, which is \\(^9C_4 = 126\\) ways.
- Exclude both R and L: Choose 4 people from the remaining 8, which is \\(^8C_4 = 70\\) ways.
Summing the first two scenarios and subtracting the third gives: \\(126 + 126 - 70 = 182\\) ways.
All methods confirm that the number of ways is 182.
