Suppose we want to choose 5 objects, without replacement, from 10 distinct objects.(If necessary, consult a list of formulas.)(a) How many ways can this be done, if the order of the choices matters?(b) How many ways can this be done, if the order of the choices does not matter?
Question
Suppose we want to choose 5 objects, without replacement, from 10 distinct objects.(If necessary, consult a list of formulas.)(a) How many ways can this be done, if the order of the choices matters?(b) How many ways can this be done, if the order of the choices does not matter?
Solution
(a) If the order of the choices matters, we use the formula for permutations of n distinct objects taken r at a time. The formula is P(n, r) = n! / (n-r)!. Here, n is the total number of objects (10) and r is the number of objects we are choosing (5). So, P(10, 5) = 10! / (10-5)! = 109876 = 30,240.
(b) If the order of the choices does not matter, we use the formula for combinations of n distinct objects taken r at a time. The formula is C(n, r) = n! / [r!(n-r)!]. Here, n is the total number of objects (10) and r is the number of objects we are choosing (5). So, C(10, 5) = 10! / [5!(10-5)!] = 252.
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