How many ways can 3 red, 2 blue, and 1 green balls be arranged in a line?

How many ways can 3 red, 2 blue, and 1 green balls be arranged in a line?

Correct Answer: 60

Step 1: Count the total number of balls. We have 3 red, 2 blue, and 1 green ball. So, total balls = 3 + 2 + 1 = 6.
Step 2: Understand that we need to arrange these 6 balls in a line.
Step 3: If all balls were different, the number of arrangements would be 6! (which means 6 factorial). This is calculated as 6 x 5 x 4 x 3 x 2 x 1 = 720.
Step 4: Since some balls are the same (3 red and 2 blue), we need to divide by the factorial of the number of identical balls to avoid counting the same arrangement multiple times.
Step 5: Calculate the arrangements for the identical balls: 3! for red balls (3 x 2 x 1 = 6) and 2! for blue balls (2 x 1 = 2) and 1! for the green ball (which is just 1).
Step 6: Now, use the formula: Total arrangements = 6! / (3! * 2! * 1!).
Step 7: Substitute the values: Total arrangements = 720 / (6 * 2 * 1) = 720 / 12 = 60.
Step 8: Therefore, the total number of ways to arrange the balls is 60.

Permutations of Multisets – The question tests the ability to calculate the number of distinct arrangements of items where some items are identical.