How many ways are there to put the balls into the boxes?
How many ways are there to place the balls into the boxes? There are only six ways.
How many ways can you distribute 4 balls into 5 boxes?
So we take the 40320 ways to areange all 8 elements then divide by the 24 ways to arrange the 4 bars, then divide again by the 24 ways to arrange the 4 dots and we get 70 ways. Much smaller than the 625 (5⁴) ways you can put 4 unique balls into 5 boxes.
How many ways can you distribute a ball into m distinct boxes?
However, the answer to the number of ways to distribute n distinct balls into n distinct boxes is trivial: each ball has a choice of m boxes, so the total number of ways to distribute n distinct balls into m distinct boxes is m n. The formula for n identical boxes in m distinct boxes is beautiful in a cute, kludgy sense.
How do you label M-1 distinct objects?
Label the space between the first two balls 1, between the second and third balls 2, between the third and fourth balls 3, and so on so that the final space between the (n-1)st ball and the mth ball is labeled m-1. Then apply an algorithm for choosing m-1 distinct objects out of n-1 objects: the number of possible choices are ( n − 1 m − 1).
How many permutations are there for each distribution of the balls?
In other words, the number of permutations corresponding to each distribution of the unlabelled balls is not constant: it depends on the distribution. Here the count of 2 3 sequences counts two of the possible distributions correctly, but it overcounts the other two by a factor of 3.
How do you calculate the cardinality of a distribution?
Then (n+m-1)C (m-1) = (n+m-1)Cn counts the distribution cardinality. This is the stars and bars approach. If the boxes are identical then distinct amounts in each box leads to overcounting by a factor of m!. If at least two boxes contain identical numbers of balls, then you are overcounting by a multinomial coefficient factor.