Suppose we want to choose 4 objects, without replacement, from 18 distinct objects. \[ \begin{array}{l}\text { (a) If the order of the choices does not matter, how many ways can this be done? } \\ \text { (b) If the order of the choices matters, how many ways can this be done? } \\ \text { ( }\end{array} \]

Let's solve each part of the problem step by step. **Given:** - Total distinct objects = 18 - Number of objects to choose = 4 - Selection is without replacement. ### Part (a): Order Does Not Matter When the order of selection **does not** matter, we're dealing with **combinations**. The number of ways to choose 4 objects from 18 is given by the combination formula: \[ \binom{18}{4} = \frac{18!}{4!(18 - 4)!} \] **Simplifying the Calculation:** 1. **Calculate the factorial terms:** \[ \binom{18}{4} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2 \times 1} \] 2. **Compute the numerator:** \[ 18 \times 17 = 306 \\ 306 \times 16 = 4,896 \\ 4,896 \times 15 = 73,440 \] 3. **Compute the denominator:** \[ 4 \times 3 \times 2 \times 1 = 24 \] 4. **Divide the numerator by the denominator:** \[ \frac{73,440}{24} = 3,060 \] **Answer to Part (a):** \[ \boxed{3,\!060\ \text{ways}} \] ### Part (b): Order Matters When the order of selection **does** matter, we're dealing with **permutations**. The number of ways to arrange 4 objects out of 18 is given by the permutation formula: \[ P(18, 4) = 18 \times 17 \times 16 \times 15 \] **Calculating the Permutation:** 1. **Multiply the terms sequentially:** \[ 18 \times 17 = 306 \\ 306 \times 16 = 4,896 \\ 4,896 \times 15 = 73,440 \] **Answer to Part (b):** \[ \boxed{73,\!440\ \text{ways}} \] ### Summary: - **(a)** There are **3,060** ways to choose 4 objects when order does not matter. - **(b)** There are **73,440** ways to choose and arrange 4 objects when order matters.