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 3 flags from a set of 6 different flags, we will calculate the number of arrangements for exactly 3, 4, 5, and 6 flags, and then sum these values.

### Step-by-Step Solution:

1. **Calculate arrangements using exactly 3 flags:**
- To choose 3 flags from 6, we can arrange them in \( P(6, 3) \) ways.
- The formula for permutations is given by:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
- For \( n = 6 \) and \( r = 3 \):
\[ P(6, 3) = \frac{6!}{(6-3)!} = \frac{6!}{3!} = \frac{6 \times 5 \times 4}{1} = 120 \]

2. **Calculate arrangements using exactly 4 flags:**
- To choose 4 flags from 6, we can arrange them in \( P(6, 4) \) ways.
- For \( n = 6 \) and \( r = 4 \):
\[ P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = \frac{6 \times 5 \times 4 \times 3}{1} = 360 \]

3. **Calculate arrangements using exactly 5 flags:**
- To choose 5 flags from 6, we can arrange them in \( P(6, 5) \) ways.
- For \( n = 6 \) and \( r = 5 \):
\[ P(6, 5) = \frac{6!}{(6-5)!} = \frac{6!}{1!} = 6 \times 5 \times 4 \times 3 \times 2 = 720 \]

4. **Calculate arrangements using all 6 flags:**
- To arrange all 6 flags, we can do this in \( P(6, 6) \) ways.
- For \( n = 6 \) and \( r = 6 \):
\[ P(6, 6) = 6! = 720 \]

5. **Sum all the arrangements:**
- Now, we add the number of arrangements for exactly 3, 4, 5, and 6 flags:
\[ \text{Total} = P(6, 3) + P(6, 4) + P(6, 5) + P(6, 6) \]
\[ \text{Total} = 120 + 360 + 720 + 720 = 1920 \]

### Final Answer:
The total number of different signals that can be made using at least 3 flags is **1920**.