Question

• A. The quantity on the left is greater
• B. The quantity on the right is greater
• C. Both are equal
• D. The relationship cannot be determined without further information

Hint:

For combination problems with conditions: - \textbf{Including} particular items: Reduce both the total number of items to choose from AND the number of items you need to choose. - \textbf{Excluding} particular items: Reduce only the total number of items to choose from.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves combinations with specific conditions: one case requires certain players to be *included*, while the other requires them to be *excluded*. These conditions change the number of players we need to select and the pool of players available for selection.
Step 2: Key Formula or Approach:
The number of ways to choose \(r\) items from a set of \(n\) is \(C(n, r) = \frac{n!}{r!(n-r)!}\).
A useful identity is \(C(n, r) = C(n, n-r)\).
Step 3: Detailed Explanation:
For Column A: Including 2 particular players
- We need to select a team of 11 players. - Since 2 specific players must be on the team, they are already chosen. - We now need to select the remaining \(11 - 2 = 9\) players. - The pool of available players is also reduced, as those 2 are no longer available for selection. We have \(16 - 2 = 14\) players left to choose from. - The number of ways is to choose 9 players from the remaining 14: \[ C(14, 9) = C(14, 14-9) = C(14, 5) = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 14 \times 13 \times 11 = 2002 \] So, Quantity A is 2002.
For Column B: Excluding 2 particular players
- We need to select a team of 11 players. - Since 2 specific players must not be on the team, we remove them from the pool of available players. - The pool of available players is now \(16 - 2 = 14\). - We need to select all 11 players for the team from this reduced pool. - The number of ways is to choose 11 players from the 14 available: \[ C(14, 11) = C(14, 14-11) = C(14, 3) = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 14 \times 13 \times 2 = 364 \] So, Quantity B is 364.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 2002
Quantity B = 364
Quantity A is greater than Quantity B.