Introduction / Context:
This question involves combinations, where we are selecting a group of players and the order of selection does not matter. The coach needs to form a starting lineup of 5 players from a squad of 12.
Given Data / Assumptions:
- Total players available = 12.
- Number of starters to be chosen = 5.
- Order of players in the list of starters does not matter; only the set of players is important.
Concept / Approach:
Since the order of selection does not matter, we use the combination formula nCr, which counts the number of ways to choose r objects from n distinct objects without regard to order. Here, n = 12 and r = 5.
Step-by-Step Solution:
Step 1: Use the combination formula: nCr = n! / (r! * (n - r)!).
Step 2: Substitute n = 12 and r = 5.
Step 3: Compute 12C5 = 12! / (5! * 7!).
Step 4: Simplify by expanding the top 5 factors: 12C5 = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1).
Step 5: Compute numerator: 12 * 11 = 132, 132 * 10 = 1320, 1320 * 9 = 11880, 11880 * 8 = 95040.
Step 6: Compute denominator: 5! = 120.
Step 7: Divide 95040 by 120 to obtain 792.
Verification / Alternative check:
You can check the arithmetic by simplifying before multiplying fully. For example, 10 / 5 = 2, 12 / 4 = 3, so we can compute (3 * 11 * 2 * 9 * 8) / (3 * 2 * 1) and obtain the same final result of 792, confirming the correctness of the combination calculation.
Why Other Options Are Wrong:
Values like 495 and 720 are common for related combinations (for example, 12C4 or 6! values) but do not match 12C5. 624 does not correspond to any standard selection count for this context.
Common Pitfalls:
A frequent mistake is to use permutations instead of combinations, which multiplies by unnecessary factors related to ordering. Another issue is miscalculating the factorial values or failing to simplify fractions properly before dividing.
Final Answer:
The number of different ways to choose 5 starters from 12 players is
792.
Discussion & Comments