What is the number of signals that can be sent by 6 flags of different colours taking one or more at a time?

Options

A. 21
B. 6
C. 720
D. 1956

Answer

To solve the problem of finding the number of signals that can be sent using 6 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 6 flags of different colors, and we want to find out how many different signals can be formed by using one or more flags at a time.

2. **Calculating Signals for Different Numbers of Flags**:
We will calculate the number of arrangements (signals) for using 1 flag, 2 flags, up to all 6 flags.

- **Using 1 Flag**:
The number of ways to choose and arrange 1 flag from 6 is given by:
\[ P(6, 1) = 6! / (6-1)! = 6! / 5! = 6 \]

- **Using 2 Flags**:
The number of ways to choose and arrange 2 flags from 6 is given by:
\[ P(6, 2) = 6! / (6-2)! = 6! / 4! = 6 \times 5 = 30 \]

- **Using 3 Flags**:
The number of ways to choose and arrange 3 flags from 6 is given by:
\[ P(6, 3) = 6! / (6-3)! = 6! / 3! = 6 \times 5 \times 4 = 120 \]

- **Using 4 Flags**:
The number of ways to choose and arrange 4 flags from 6 is given by:
\[ P(6, 4) = 6! / (6-4)! = 6! / 2! = 6 \times 5 \times 4 \times 3 = 360 \]

- **Using 5 Flags**:
The number of ways to choose and arrange 5 flags from 6 is given by:
\[ P(6, 5) = 6! / (6-5)! = 6! / 1! = 6 \times 5 \times 4 \times 3 \times 2 = 720 \]

- **Using 6 Flags**:
The number of ways to arrange all 6 flags is given by:
\[ P(6, 6) = 6! / (6-6)! = 6! / 0! = 720 \]

3. **Summing All Possible Arrangements**:
Now, we will sum all the arrangements calculated above:
\[ \text{Total Signals} = P(6, 1) + P(6, 2) + P(6, 3) + P(6, 4) + P(6, 5) + P(6, 6) \]
\[ = 6 + 30 + 120 + 360 + 720 + 720 = 1956 \]

4. **Conclusion**:
Therefore, the total number of signals that can be sent using 6 flags of different colors, taking one or more at a time, is **1956**.