In how many ways 20 identical balls can be put into 5 distinct bins such that each bin contains at least 2 balls? Firstly, I have to find the total no of ways in which 20 identical balls can be put into 5 distinct bins.
Then, I have to subtract the no of cases when no of balls in at least one bin is less than 2.
But, I'm not getting how to find that no.
If possible, plz suggest me various ways to approach and solve this kind of problems.
Plz help me with the complete solution to this problem.
As the balls the identical, I think we can just start the process with each box having two balls in each box. The identical nature of the balls tells us it can be done in only 1 way. So we are left with the task of filling 10 identical balls in 5 boxes.

The answer 5^20 is not right for identical balls, think of a less severe situation with 4 balls and 2 boxes, the problem is essentially asking in how many ways can you put the 4 balls in two different containers.
Let's say the balls are b1,b2,b3 and b4 and say one 1 way of filling it is,

b1,b2 goes to container A
b3,b4 goes to container B

another way of filling it proposes

b1,b3 goes to container A
b2,b4 goes to container B

But in our case b1=b2=b3=b4 thereby both the cases are equivalent.
I'm fact, there are 4C2 equivalent ways of filling the container 2 balls at a time.
Same goes to the case of the original problem of filling 5 container with 20 identical balls, so there will be less number of possibilities.
Firstly give two balls to each box then use stick method to assign the remaining balls.
The answer is 11C4+2(11C3)+11C2+11C1=726
Ask me if you don't get this!