A tricolour flag is to be formed having three adjacent strips of three different colours choosen from 6 different colours. How many different coloured flags can be formed with different design in which allthe three different coloured flags can be formed with different design in which all the three strips are always in horizontal positions ?

Answer

To solve the problem of how many different colored flags can be formed with three adjacent strips of three different colors chosen from six different colors, we can follow these steps:

### Step 1: Choose the Colors for the Strips
We need to select 3 different colors from a total of 6 colors. The number of ways to choose 3 colors from 6 is given by the combination formula:

\[ \text{Number of ways to choose 3 colors} = \binom{6}{3} \]

Calculating this:

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 2: Arrange the Chosen Colors
Once we have chosen 3 colors, we need to arrange them in the 3 strips. The number of ways to arrange 3 colors is given by the factorial of the number of colors:

\[ \text{Number of arrangements of 3 colors} = 3! = 6 \]

### Step 3: Calculate the Total Number of Flags
Now, we multiply the number of ways to choose the colors by the number of arrangements:

\[ \text{Total number of flags} = \text{Number of ways to choose colors} \times \text{Number of arrangements} \]

Substituting the values we calculated:

\[ \text{Total number of flags} = 20 \times 6 = 120 \]

Thus, the total number of different colored flags that can be formed is **120**.

### Summary of Steps:
1. Calculate the number of ways to choose 3 colors from 6: \( \binom{6}{3} = 20 \).
2. Calculate the number of arrangements of the 3 chosen colors: \( 3! = 6 \).
3. Multiply the results from steps 1 and 2 to find the total number of flags: \( 20 \times 6 = 120 \).