Suppose we want to choose 7 letters, without replacement, from 11 distinct letters.(If necessary, consult a list of formulas.)(a) If the order of the choices is not relevant, how many ways can this be done?(b) If the order of the choices is relevant, how many ways can this be done?
Question
Suppose we want to choose 7 letters, without replacement, from 11 distinct letters.(If necessary, consult a list of formulas.)(a) If the order of the choices is not relevant, how many ways can this be done?(b) If the order of the choices is relevant, how many ways can this be done?
Solution
(a) If the order of the choices is not relevant, we use the combination formula. The number of ways to choose 7 letters from 11 is given by "11 choose 7", which is calculated as:
C(n, k) = n! / [k!(n-k)!]
where n is the total number of options (11 in this case), k is the number of choices we make (7 in this case), and "!" denotes factorial, which is the product of all positive integers up to that number.
So, "11 choose 7" = 11! / [7!(11-7)!] = 330 ways.
(b) If the order of the choices is relevant, we use the permutation formula. The number of ways to choose 7 letters from 11, where the order matters, is given by "11 permute 7", which is calculated as:
P(n, k) = n! / (n-k)!
So, "11 permute 7" = 11! / (11-7)! = 1,977,600 ways.
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