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