Question

Use these formulas to help you answer the questions.
Permutation formula: P(n, r) = n!(n - r)! Combination formula: C(n, r) = n!r!(n - r)!


Adam has been given a list of 8 bands and asked to place a vote. How many different votes are possible for the top three bands if order is important? (12 points; 2 per ? mark)

P(n, r) = n!(n - r)! → ? !(? - ?)! = ? ? = ?

Answer 1

To determine the number of different votes for the top three bands when the order is important, you can use the permutation formula:

\[ P(n, r) = \frac{n!}{(n - r)!} \]

In this scenario:
- \( n \) is the total number of bands, which is 8.
- \( r \) is the number of top positions (or bands being voted for), which is 3.

Now, substituting into the permutation formula:

### Step 1: Identify the values
- \( n = 8 \)
- \( r = 3 \)

### Step 2: Plug the values into the formula
\[ P(8, 3) = \frac{8!}{(8 - 3)!} = \frac{8!}{5!} \]

### Step 3: Simplify
To simplify \( \frac{8!}{5!} \), we can expand \( 8! \):
\[ 8! = 8 \times 7 \times 6 \times 5! \]
So,
\[ P(8, 3) = \frac{8 \times 7 \times 6 \times 5!}{5!} \]

The \( 5! \) cancels out:
\[ P(8, 3) = 8 \times 7 \times 6 \]

### Step 4: Calculate the result
Now, we compute:
\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]

Thus, the total number of different votes possible for the top three bands is **336**.

### Fill in the blanks
Now, filling in the blanks of the provided format:
\[ P(n, r) = n!(n - r)! \]
→ \( 8!(8 - 3)! = 8!5! = 336 \)

So the answers are as follows:
1. \( n = 8 \)
2. \( r = 3 \)
3. \( P(8, 3) = 336 \)