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

Question

UpStudy AI · Accepted Answer

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.
There are 3,060 ways when order doesn't matter and 73,440 ways when order does matter.

Suppose we want to choose 4 objects, without replacement, from 18 distinct objects.

Upstudy AI Solution

Answer

Solution

Part (a): Order Does Not Matter

Part (b): Order Matters

Summary:

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