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? \( \square \)

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.