What is the number of ways in which n identical objects can be divided into r groups where each group can have any number of objects?

Originally Answered: Permutation and Combination: Number of ways in which n identical objects can be divided into r groups where each group can have any number of objects including 0? This is called the Bose-Einstein coefficient (because of a connection with a bizarre state of matter known as Bose-Einstein condensate).

What is the number of ways in which n identical objects can be divided into r groups where each group can have any number of objects excluding 0?

Number of ways exactly 1 object can be placed in r groups = nCr. Number of remaining objects = (n-r) Now, each of these (n-r) objects can be placed into any of the r groups. Hence, number of ways of distributing them = (n-r)^r.

Why N objects have more possible permutations than combinations?

In this case, “order doesn’t matter,” and the different ways are different combinations. In the calculation of “how many (permutations),(combinations) can be made from K objects out of N candidates, there will be more permutations than combinations, because each combination can be rearranged to make many permutations.

How many ways are there to place 6 identical objects into 3 different bins?

So left 3 balls can be put in C(5,2). But, the answer is 540.

How many ways are there to put 4 balls into 4 boxes where the balls are identical and the boxes are also identical?

Thus in total the balls can be divided in 15 ways in the four indistinguishable boxes.

How many ways can you divide a group into two?

Suppose x is a particular element of the set. The two groups are completely determined by choosing which of the remaining N−1 elements are in the same subset as x. There are 2N−1 ways to choose a subset of the remaining N−1 elements, so there are 2N−1 ways of dividing the set into two groups.