If out of 6 flags any number of flags can be shown at a time find how many different signals can be made out of them.

Answer

To solve the problem of finding how many different signals can be made using 6 flags where any number of flags can be shown at a time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 6 flags, and we can use any number of them to create signals. The arrangement of flags matters, meaning that different orders of the same flags count as different signals.

2. **Using Permutations**:
The number of ways to arrange \( r \) flags out of \( n \) flags is given by the formula for permutations:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
where \( n \) is the total number of flags and \( r \) is the number of flags chosen.

3. **Considering All Possible Cases**:
We can use 1 flag, 2 flags, 3 flags, 4 flags, 5 flags, or all 6 flags. Therefore, we need to calculate the permutations for each case:
- For \( r = 1 \): \( P(6, 1) = \frac{6!}{(6-1)!} = \frac{6!}{5!} = 6 \)
- For \( r = 2 \): \( P(6, 2) = \frac{6!}{(6-2)!} = \frac{6!}{4!} = 6 \times 5 = 30 \)
- For \( r = 3 \): \( P(6, 3) = \frac{6!}{(6-3)!} = \frac{6!}{3!} = 6 \times 5 \times 4 = 120 \)
- For \( r = 4 \): \( P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = 6 \times 5 \times 4 \times 3 = 360 \)
- For \( r = 5 \): \( P(6, 5) = \frac{6!}{(6-5)!} = \frac{6!}{1!} = 6 \times 5 \times 4 \times 3 \times 2 = 720 \)
- For \( r = 6 \): \( P(6, 6) = \frac{6!}{(6-6)!} = \frac{6!}{0!} = 6! = 720 \)

4. **Summing All Cases**:
Now, we sum all the permutations calculated:
\[ \text{Total Signals} = P(6, 1) + P(6, 2) + P(6, 3) + P(6, 4) + P(6, 5) + P(6, 6) \]
\[ = 6 + 30 + 120 + 360 + 720 + 720 \]
\[ = 6 + 30 = 36 \]
\[ = 36 + 120 = 156 \]
\[ = 156 + 360 = 516 \]
\[ = 516 + 720 = 1236 \]
\[ = 1236 + 720 = 1956 \]

5. **Final Answer**:
The total number of different signals that can be made using the 6 flags is **1956**.