Question

A cricket team of 11 players from 16 players is to be selected. If three particular players are always included in the team, then the number of ways of selecting the team is

• A. 1287
• B. 1187
• C. 1117
• D. 1298
• E. 1349

Hint:

Logic Tip: In "always included" problems, subtract the fixed items from BOTH the total pool ($n$) and the selection size ($r$). In "never included" problems, subtract the excluded items ONLY from the total pool ($n$).

Answer 1

The Correct Option is A

Concept:
The number of ways to choose $r$ items from a set of $n$ items without regarding the order is given by the combination formula: $$^{n}C_{r} = \frac{n!}{r!(n-r)!}$$
Step 1: Adjust the pool of players based on the conditions.
Total available players = 16. Total required team size = 11. Since 3 specific players are *always* included, those 3 spots are effectively filled, and those 3 players are removed from the selection pool.
Step 2: Determine the new selection targets.
Remaining players to choose from: $n = 16 - 3 = 13$. Remaining spots to fill on the team: $r = 11 - 3 = 8$. We now need to select 8 players from the remaining pool of 13 players.
Step 3: Calculate the combinations.
Apply the combination formula for choosing 8 from 13: $$^{13}C_{8} = \frac{13!}{8! \cdot (13-8)!} = \frac{13!}{8! \cdot 5!}$$ Expand the factorials to cancel terms: $$= \frac{13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8!}{8! \cdot (5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$$ $$= \frac{13 \cdot 12 \cdot 11 \cdot 10 \cdot 9}{120}$$ $$= 13 \cdot 11 \cdot 9$$ $$= 1287$$