Arranging Balls in a Row

The problem states that there are three blue balls, four red balls, and five green balls. We need to find the number of ways these balls can be arranged in a row.

Solution:

To solve this problem, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order. In this case, we want to arrange the balls in a row, so the order matters.

Step 1: Determine the total number of balls

The total number of balls is the sum of the number of blue, red, and green balls.
Total number of balls = 3 (blue) + 4 (red) + 5 (green) = 12 balls

Step 2: Calculate the number of ways to arrange the balls

Since the order matters, we can use the concept of permutations to calculate the number of ways to arrange the balls. The formula for permutations is:

nPr = n! / (n - r)!

Where n is the total number of objects and r is the number of objects to be arranged.

In this case, we have 12 balls in total, so n = 12. We want to arrange all 12 balls in a row, so r = 12.

Using the formula, we can calculate the number of ways to arrange the balls:

12P12 = 12! / (12 - 12)! = 12!

The factorial of 12 is the product of all positive integers from 1 to 12:

12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Calculating the factorial of 12 gives us:

12! = 479,001,600

Therefore, there are 479,001,600 ways to arrange the balls in a row.

Step 3: Conclusion

In conclusion, there are 479,001,600 ways to arrange the three blue balls, four red balls, and five green balls in a row.