Suppose we want to choose 4 letters, without replacement, from 17 distinct letters. (If necessary, consult a list of formulas.) \[ \text { (a) If the order of the choices is relevant, how many ways can this be done? } \] (b) If the order of the choices is not relevant, how many ways can this be done? \( \square \)
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To solve part (a), where order matters, we use permutations. The number of ways to choose 4 letters from 17 distinct letters is given by the formula for permutations: \[ P(n, r) = \frac{n!}{(n - r)!} \] Here, \(n\) is 17 and \(r\) is 4. Thus, \[ P(17, 4) = \frac{17!}{(17-4)!} = \frac{17!}{13!} = 17 \times 16 \times 15 \times 14 = 57120. \] So, there are 57,120 ways to choose 4 letters if the order of the choices is relevant. For part (b), where order does not matter, we use combinations. The number of ways to choose 4 letters from 17 is given by the formula for combinations: \[ C(n, r) = \frac{n!}{r! (n - r)!} \] Thus, \[ C(17, 4) = \frac{17!}{4! \cdot 13!} = \frac{17 \times 16 \times 15 \times 14}{4 \times 3 \times 2 \times 1} = \frac{57120}{24} = 2380. \] Therefore, there are 2,380 ways to choose 4 letters without caring about the order.