Read the information and answer the questions given below:
A man can send a massage by using six flags. He had 4 black flags. 3 blue flags and 1 green flag.
In how many ways a massage can be transmitted?

Options

A. 108
B. 185
C. 28
D. 136

Answer

To solve the problem of how many ways a message can be transmitted using the flags, we need to consider the different combinations of flags available. The man has:

- 4 black flags
- 3 blue flags
- 1 green flag

The total number of flags he can use is 6 (4 black + 3 blue + 1 green). We will consider different cases based on how many flags of each color are used.

### Step 1: Identify the cases based on flag usage

1. **Case 1**: Using 4 black flags and 2 flags of other colors (either blue or green).
2. **Case 2**: Using 3 blue flags and 3 flags of other colors (black or green).
3. **Case 3**: Using 1 green flag and the rest can be black or blue.

### Step 2: Calculate the number of arrangements for each case

#### Case 1: 4 Black Flags
- **Sub-case 1**: 4 black flags + 1 blue + 1 green
- Total flags = 6 (4B, 1G, 1B)
- Arrangements = 6! / (4! * 1! * 1!) = 30 ways

- **Sub-case 2**: 4 black flags + 2 blue
- Total flags = 6 (4B, 2B)
- Arrangements = 6! / (4! * 2!) = 15 ways

Total for Case 1 = 30 + 15 = 45 ways

#### Case 2: 3 Blue Flags
- **Sub-case 1**: 3 blue + 3 black
- Total flags = 6 (3B, 3B)
- Arrangements = 6! / (3! * 3!) = 20 ways

- **Sub-case 2**: 3 blue + 2 black + 1 green
- Total flags = 6 (3B, 2B, 1G)
- Arrangements = 6! / (3! * 2! * 1!) = 60 ways

Total for Case 2 = 20 + 60 = 80 ways

#### Case 3: 1 Green Flag
- **Sub-case 1**: 1 green + 3 black + 2 blue
- Total flags = 6 (1G, 3B, 2B)
- Arrangements = 6! / (1! * 3! * 2!) = 60 ways

Total for Case 3 = 60 ways

### Step 3: Sum all the ways from each case

Total number of ways = Case 1 + Case 2 + Case 3
= 45 + 80 + 60 = 185 ways

### Final Answer
Thus, the total number of ways the message can be transmitted using the flags is **185**.

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