Suppose we want to choose 6 letters, without replacement, from 14 distinct letters.(If necessary, consult a list of formulas.)(a) If the order of the choices is not taken into consideration, how many ways can this be done?(b) If the order of the choices is taken into consideration, how many ways can this be done?
Question
Suppose we want to choose 6 letters, without replacement, from 14 distinct letters.(If necessary, consult a list of formulas.)(a) If the order of the choices is not taken into consideration, how many ways can this be done?(b) If the order of the choices is taken into consideration, how many ways can this be done?
Solution
(a) If the order of the choices is not taken into consideration, the number of ways to choose 6 letters from 14 distinct letters is given by the combination formula, which is C(n, r) = n! / [(n-r)! * r!], where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
Here, n = 14 (the total number of distinct letters) and r = 6 (the number of letters to choose).
So, C(14, 6) = 14! / [(14-6)! * 6!] = 3003 ways.
(b) If the order of the choices is taken into consideration, the number of ways to choose 6 letters from 14 distinct letters is given by the permutation formula, which is P(n, r) = n! / (n-r)!, where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
Here, n = 14 (the total number of distinct letters) and r = 6 (the number of letters to choose).
So, P(14, 6) = 14! / (14-6)! = 752,538 ways.
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