The number of signals that can be sent by 5 flags of different colours, taking one or more at a time is :

Options

A. 63
B. 1956
C. 720
D. 21

Answer

To solve the problem of finding the number of signals that can be sent by 5 flags of different colors, taking one or more at a time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 flags of different colors. We can use any combination of these flags to send signals, taking one or more flags at a time.

2. **Calculating the Combinations**: We need to consider the different cases based on how many flags we choose:
- Case 1: Choosing 1 flag from 5 flags.
- Case 2: Choosing 2 flags from 5 flags.
- Case 3: Choosing 3 flags from 5 flags.
- Case 4: Choosing 4 flags from 5 flags.
- Case 5: Choosing all 5 flags.

3. **Using Combinations and Permutations**:
- For each case, we will calculate the number of ways to choose the flags and then arrange them.
- The formula for combinations is given by \( nCr = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose.
- The number of arrangements (permutations) of \( r \) items is given by \( r! \).

4. **Calculating Each Case**:
- **Case 1**: Choosing 1 flag:
\[ \text{Ways} = 5C1 \times 1! = 5 \times 1 = 5 \]
- **Case 2**: Choosing 2 flags:
\[ \text{Ways} = 5C2 \times 2! = 10 \times 2 = 20 \]
- **Case 3**: Choosing 3 flags:
\[ \text{Ways} = 5C3 \times 3! = 10 \times 6 = 60 \]
- **Case 4**: Choosing 4 flags:
\[ \text{Ways} = 5C4 \times 4! = 5 \times 24 = 120 \]
- **Case 5**: Choosing all 5 flags:
\[ \text{Ways} = 5C5 \times 5! = 1 \times 120 = 120 \]

5. **Adding All Cases Together**:
\[ \text{Total Signals} = 5 + 20 + 60 + 120 + 120 = 325 \]

### Final Answer:
The total number of signals that can be sent by 5 flags of different colors, taking one or more at a time, is **325**.