Final Answer:
a) Permutations P(6, 3) = 120: This represents the number of possible orderings for choosing 3 distinct objects out of 6.
b) Combinations C(6, 3) = 20: This represents the number of distinct choices you can make by picking 3 objects out of 6, without considering their order.
c) Factorials F(6!) = 720: This represents the total number of arrangements for all 6 objects, regardless of choosing any specific group.
Step-by-step explanation:
a) Permutations: Order matters here. We use the formula:
P(n, r) = n! / (n - r)!
P(6, 3) = 6! / (6 - 3)! = 6 * 5 * 4 = 120
b) Combinations: Order doesn't matter. We use the formula:
C(n, r) = n! / (r! * (n - r)!)
C(6, 3) = 6! / (3! * (6 - 3)!) = (6 * 5 * 4) / (3 * 2 * 1) = 20
c) Factorials: This calculates all possible arrangements for any group size within the 6 objects. We use the formula:
F(n) = n!
F(6!) = 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Remember, permutations consider order, combinations don't, and factorials consider all possible arrangements for any group size.
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Complete Question
Calculate the following: a) Permutations P(6, 3). b) Combinations C(6, 3). c) Factorials F(6!)
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