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

Options

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

Explanation

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 break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 6 flags of different colors and we need to find the total number of signals that can be formed using one or more flags.

2. **Using Combinations and Permutations**:
For each possible number of flags (from 1 to 6), we will calculate the number of ways to choose the flags and then arrange them.

3. **Calculating for 1 Flag**:
- The number of signals using 1 flag is simply the number of flags available.
- **Calculation**:
\[ \text{Signals with 1 flag} = 6 \]

4. **Calculating for 2 Flags**:
- We need to choose 2 flags from 6 and arrange them.
- **Formula**:
\[ \text{Signals with 2 flags} = \binom{6}{2} \times 2! = \frac{6!}{(6-2)! \cdot 2!} \times 2! = \frac{6 \times 5}{1} = 30 \]

5. **Calculating for 3 Flags**:
- We need to choose 3 flags from 6 and arrange them.
- **Formula**:
\[ \text{Signals with 3 flags} = \binom{6}{3} \times 3! = \frac{6!}{(6-3)! \cdot 3!} \times 6 = \frac{6 \times 5 \times 4}{1 \times 2 \times 3} \times 6 = 120 \]

6. **Calculating for 4 Flags**:
- We need to choose 4 flags from 6 and arrange them.
- **Formula**:
\[ \text{Signals with 4 flags} = \binom{6}{4} \times 4! = \frac{6!}{(6-4)! \cdot 4!} \times 24 = \frac{6 \times 5}{1 \times 2} \times 24 = 360 \]

7. **Calculating for 5 Flags**:
- We need to choose 5 flags from 6 and arrange them.
- **Formula**:
\[ \text{Signals with 5 flags} = \binom{6}{5} \times 5! = \frac{6!}{(6-5)! \cdot 5!} \times 120 = 6 \times 120 = 720 \]

8. **Calculating for 6 Flags**:
- We need to use all 6 flags and arrange them.
- **Formula**:
\[ \text{Signals with 6 flags} = \binom{6}{6} \times 6! = 1 \times 720 = 720 \]

9. **Total Signals**:
- Now, we add all the signals calculated above:
\[ \text{Total Signals} = 6 + 30 + 120 + 360 + 720 + 720 = 1956 \]

### Final Answer:
The total number of signals that can be sent using 6 flags of different colors, taking one or more at a time, is **1956**.