Counting Tricolor Flags from Six Colors and Their Arrangements

Imagine you have a set of 18 strips in six different colors: blue, red, green, yellow, purple, and orange. Each color appears exactly three times. You want to know how many different tricolor flags you can make using these strips, considering different scenarios regarding the choice of colors.

1. Choosing 3 Stripes

Let's break down the problem into two main cases: when the three chosen colors are all different and when some of the colors are the same.

1.1 All Different Colors

In this scenario, we need to select 3 different colors from the 6 available. The number of ways to do this is given by the combination formula:

Binomial Coefficient: ({n choose k} frac{n!}{k!(n-k)!})

Here, (n 6) (total number of colors) and (k 3) (number of colors to choose).

({6 choose 3} frac{6!}{3!3!} frac{6 times 5 times 4}{3 times 2 times 1} 20)

Once we have selected 3 different colors, we can arrange them in (3!) (3 factorial) different ways.

(3! 3 times 2 times 1 6)

Therefore, the total number of different flags with all different colors is:

20 (times) 6 120

1.2 Two Colors the Same, One Different

In this case, we choose 1 color to appear twice and a different color to appear once.

1. Select the color that appears twice: There are 6 options (one for each color).

2. Select the color that appears once: After choosing the first color, there are 5 colors left to choose from.

The number of ways to choose the colors is:

6 (for the color that appears twice) (times) 5 (for the color that appears once) 30

Next, we need to arrange these colors. The arrangement of two identical colors and one different color can be calculated using the formula for permutations of a multiset:

(frac{n!}{n_1! cdot n_2!})

Here, (n 3) (total number of colors), (n_1 2) (the count of the first color), and (n_2 1) (the count of the second color).

(frac{3!}{2! cdot 1!} frac{3 times 2 times 1}{2 times 1} 3)

Therefore, the total number of different flags with two colors the same and one different is:

30 (times) 3 90

3. Total Different Flags

Finally, we add both cases together to find the total number of different tricolor flags:

120 (all different) 90 (two the same, one different) 210

Hence, the total number of different tricolor flags that can be made is 210.

By understanding these combinations and permutations, you can explore the full range of possibilities for creating unique tricolor flags from the given set of strips.