To solve the problem of how many ways a message can be transmitted using all three colored flags (4 black flags, 3 blue flags, and 1 green flag), we can break down the solution step by step.
### Step 1: Understand the Problem
We need to use all three colors of flags: black, blue, and green. The total flags available are:
- Black: 4
- Blue: 3
- Green: 1
Since we have only one green flag, it must be included in every combination.
### Step 2: Identify Possible Combinations
We can have different combinations of black and blue flags along with the green flag. The combinations can be:
1. 1 Green, 2 Black, 3 Blue
2. 1 Green, 3 Black, 2 Blue
3. 1 Green, 4 Black, 1 Blue
### Step 3: Calculate the Number of Ways for Each Combination
For each combination, we will calculate the number of ways to arrange the flags.
#### Combination 1: 1 Green, 2 Black, 3 Blue
- Total flags = 1 + 2 + 3 = 6
- The number of arrangements = \( \frac{6!}{1! \cdot 2! \cdot 3!} \)
Calculating:
\[
6! = 720
\]
\[
1! = 1, \quad 2! = 2, \quad 3! = 6
\]
\[
\text{Arrangements} = \frac{720}{1 \cdot 2 \cdot 6} = \frac{720}{12} = 60
\]
#### Combination 2: 1 Green, 3 Black, 2 Blue
- Total flags = 1 + 3 + 2 = 6
- The number of arrangements = \( \frac{6!}{1! \cdot 3! \cdot 2!} \)
Calculating:
\[
\text{Arrangements} = \frac{720}{1 \cdot 6 \cdot 2} = \frac{720}{12} = 60
\]
#### Combination 3: 1 Green, 4 Black, 1 Blue
- Total flags = 1 + 4 + 1 = 6
- The number of arrangements = \( \frac{6!}{1! \cdot 4! \cdot 1!} \)
Calculating:
\[
\text{Arrangements} = \frac{720}{1 \cdot 24 \cdot 1} = \frac{720}{24} = 30
\]
### Step 4: Sum the Arrangements
Now, we add the number of arrangements from all combinations:
\[
60 + 60 + 30 = 150
\]
### Final Answer
The total number of ways a message can be transmitted using all three colored flags is **150**.
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