Final answer:
To distribute 14 dodgeballs into 5 bins, we can consider two cases: when all 5 bins have 6 balls each, and when one bin has fewer than 6 balls. In the first case, the number of ways is 0. In the second case, the total number of ways is 236.
Step-by-step explanation:
To solve this problem, we can use permutations and combinations. We need to distribute 14 dodgeballs into 5 bins, with no bin holding more than 6 balls. Let's consider this in two cases:
Case 1: If all 5 bins have 6 balls each.
In this case, we can calculate the number of ways to distribute the remaining 14 - (5 * 6) = 14 - 30 = -16 balls into the 5 bins. Since we cannot have a negative number of balls, the number of ways in this case is 0.
Case 2: If one bin has fewer than 6 balls.
In this case, we can consider the different possibilities for the number of balls in the smallest bin: 1, 2, 3, 4, or 5.
If the smallest bin has 1 ball, we need to distribute the remaining 14 - (1 * 5) = 14 - 5 = 9 balls into the 4 remaining bins. This can be done in 9C4 = 126 ways.
Similarly, if the smallest bin has 2, 3, 4, or 5 balls, we distribute the remaining balls in 8C4, 7C4, 6C4, and 5C4 ways respectively.
Therefore, the total number of ways to distribute the dodgeballs is 0 + 126 + 70 + 35 + 5 = 236.