In a puzzle there are five boxes. There are also 25 plates with 5 different colours. Plates are numbered one to five for each coloured plates. Boxes too are numbered one to five. The puzzle is to find out the total number of arrangements possible if we can place total number of 5 plates at any time i.e. total number of plates in every time should equal to five in each arrangement. The box numbered ‘n’ (n=1..5) can have any coloured plate as long as the plate’s number is ‘n’. For e.g. the box numbered four can have all the 5 plates with colours numbered four or any colour of the plates as long as plates are numbered four. Total of 5 plates has to be placed in boxes in each arrangement. each boxes can hold any coloured plate as long as number of box and plate are same. at any time all the five colors should be in among 5 boxes. Using nCr formula works or n! works but it only answers problem where a box can accommodate only one plate and all the boxes are filled with one. I want to figure out when each box can accommodate more than one plates still total of all the plates in all the boxes are 5 (or say n for general case).

What is the general formula to find out the total number of arrangements possible in such a way?