How many different signals can be made with 6 given flags of different colours ?

Answer

To find the total number of different signals that can be made with 6 given flags of different colors, we can use the concept of permutations. We will consider the number of ways to arrange 1 flag, 2 flags, 3 flags, 4 flags, 5 flags, and all 6 flags.

### Step-by-Step Solution:

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

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

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

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

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

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

7. **Total Number of Signals:**
- Now, we add all the different arrangements together:
\[ \text{Total} = P(6, 1) + P(6, 2) + P(6, 3) + P(6, 4) + P(6, 5) + P(6, 6) \]
\[ = 6 + 30 + 120 + 360 + 720 + 720 = 1956 \]

### Final Answer:
The total number of different signals that can be made with 6 given flags of different colors is **1956**.