Final answer:
To determine the number of ways to distribute 200 similar balls into 5 different boxes with specific constraints, we use the stars and bars combinatorial method after placing the minimum number of required balls in each box.
Step-by-step explanation:
The question asks to calculate the number of ways in which 200 similar balls can be distributed into 5 different boxes with the constraint that there should be at least 2, 3, 4, 5, and 6 balls in the respective boxes. First, we follow the constraints and place the minimum required balls in each box: 2 in the first, 3 in the second, 4 in the third, 5 in the fourth, and 6 in the fifth. This leaves us with 200 - (2+3+4+5+6) = 180 balls to distribute freely.
The problem now reduces to finding the number of ways to distribute 180 indistinguishable balls into 5 distinguishable boxes, which is a problem of partitioning with distinguishable boxes. This is a combinatorial problem that can be solved using the stars and bars method (also known as balls and urns). The formula for distributing n identical items into r different groups is given by this combinations formula:
(n + r - 1) choose (r - 1) = (n + r - 1)! / [(r - 1)!n!]
By applying this formula to our situation, we take n as 180 and r as 5 to calculate the total number of ways to distribute the remaining balls.