Question

The number of ways in which a team of 11 players can be formed out of 25 players, if 6 out of them are always to be included and 5 of them are always to be excluded, is ______.

• A. 2002
• B. $^{20}\text{C}_{11}$
• C. $^{20}\text{C}_6$
• D. $^{14}\text{C}_5$

Hint:

Inclusion/Exclusion trick: If $p$ are always included and $q$ are always excluded from $n$ items to select $r$, the formula is simply $^{(n - p - q)}\text{C}_{(r - p)}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We must calculate the number of combinations to form a team under strict inclusion and exclusion conditions.

Step 2: Detailed Explanation:
Total number of players available = 25.
Total number of players required for the team = 11.
Condition 1: 6 players are always included.
Since these 6 players are already permanently placed on the team, we don't need to select them; they are a given.
Remaining spots to fill on the team = $11 - 6 = 5 spots$.
Condition 2: 5 players are always excluded.
These 5 players are removed from the available pool completely.
Let's calculate the size of the remaining pool of players we can actually choose from:
Remaining pool = Total players - (Always included) - (Always excluded)
Remaining pool = $25 - 6 - 5 = 14 players$.
So, the problem mathematically simplifies to: How many ways can we choose the remaining 5 spots from the remaining pool of 14 players?
This is a standard combination calculation:
Number of ways = $^{14}\text{C}_5$

Step 3: Final Answer:
The number of ways is $^{14}\text{C}_5$, matching option (d).