How many different signals, can be made by 5 flags from 8 flags of different colours?

Options

A. (1) 6270
B. (2) 1680
C. (3) 20160
D. (4) 6720

Answer

To solve the problem of how many different signals can be made by using 5 flags from a total of 8 flags of different colors, we will use the concept of permutations since the order in which the flags are arranged matters.

### Step-by-Step Solution:

1. **Identify the Total Flags and Flags to Choose**:
- We have a total of \( n = 8 \) flags.
- We need to choose \( r = 5 \) flags.

2. **Use the Permutation Formula**:
- The formula for permutations is given by:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
- Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \), and \( (n - r)! \) is the product of all positive integers up to \( n - r \).

3. **Substitute the Values into the Formula**:
- Substitute \( n = 8 \) and \( r = 5 \) into the formula:
\[ P(8, 5) = \frac{8!}{(8 - 5)!} = \frac{8!}{3!} \]

4. **Calculate \( 8! \) and \( 3! \)**:
- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3! \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

5. **Simplify the Expression**:
- Now, we can cancel \( 3! \) in the numerator and denominator:
\[ P(8, 5) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3!}{3!} = 8 \times 7 \times 6 \times 5 \times 4 \]

6. **Perform the Multiplication**:
- Calculate \( 8 \times 7 = 56 \)
- Next, \( 56 \times 6 = 336 \)
- Then, \( 336 \times 5 = 1680 \)
- Finally, \( 1680 \times 4 = 6720 \)

7. **Final Answer**:
- Therefore, the total number of different signals that can be made by using 5 flags from 8 flags of different colors is:
\[ \boxed{6720} \]