Using 6 different flags, how many different signals can be made by using atleast three flags, arranging on above the other ?

Answer

To solve the problem of how many different signals can be made using at least three flags from six different flags, we will break it down into cases based on the number of flags used.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the total number of arrangements of signals using at least three flags from a set of six different flags.

2. **Case 1: Using 3 Flags**:
- We can choose 3 flags from 6. The number of ways to choose 3 flags from 6 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
- The number of ways to choose 3 flags from 6 is \( \binom{6}{3} \).
- After choosing the flags, we can arrange them. The number of arrangements of 3 flags is \( 3! \).
- Thus, the total arrangements for this case is:
\[ \text{Arrangements for 3 flags} = \binom{6}{3} \times 3! = 20 \times 6 = 120 \]

3. **Case 2: Using 4 Flags**:
- The number of ways to choose 4 flags from 6 is \( \binom{6}{4} \).
- The number of arrangements of 4 flags is \( 4! \).
- Thus, the total arrangements for this case is:
\[ \text{Arrangements for 4 flags} = \binom{6}{4} \times 4! = 15 \times 24 = 360 \]

4. **Case 3: Using 5 Flags**:
- The number of ways to choose 5 flags from 6 is \( \binom{6}{5} \).
- The number of arrangements of 5 flags is \( 5! \).
- Thus, the total arrangements for this case is:
\[ \text{Arrangements for 5 flags} = \binom{6}{5} \times 5! = 6 \times 120 = 720 \]

5. **Case 4: Using 6 Flags**:
- The number of ways to choose all 6 flags is \( \binom{6}{6} \).
- The number of arrangements of 6 flags is \( 6! \).
- Thus, the total arrangements for this case is:
\[ \text{Arrangements for 6 flags} = \binom{6}{6} \times 6! = 1 \times 720 = 720 \]

6. **Total Arrangements**:
- Now, we add all the arrangements from each case:
\[ \text{Total arrangements} = 120 + 360 + 720 + 720 = 1920 \]

### Final Answer:
The total number of different signals that can be made using at least three flags is **1920**.