Distinct arrangement of n number of  objects among  r distinct cells (without ordering inside a cell)

 

   
   
Value of     n
Value of     r
Value of p things excluded/included
Each cell to contain at least q objects
Partition arrangements (maximum 5) ----
 

* q1 objects are of 1 type, q2 of another type, q3 another type & sum of (q1+q2+....qn)=n. Here we take upto 5 types in the row "Partition arrangements"

  When Objects are indistinguishable from one another  
Description Formulae  
Each cell can have one or more objects or can be empty  n+r-1Cn
Each cell can have one  object or empty (r => n) r! / [n!(r-n)!]
Each cell to have at least one object but not empty n-1Cr-1
Each cell to have at least q objects n+r-1-rqCr-1
distribution of q1,q2,q3,...qn* objects aggregating n & n<=r [r! / (q1!q2!q3!...qn)][1/(r-n)!]
Arrangement with distribution as per partition n! / (n1!n2!n3!n4!n5!)
  When Objects are distinguishable from one another  
Description Formulae  
Each cell can have one or more objects or can be empty (n <r) rn
Distribution where ordering in a cell matters (n+r-1)! /(r-1)!
Each cell can have one object or  empty  
Each cell can have at least one object but not empty  
Arrangement with distribution as per partition n! / (n1!n2!n3!n4!n5!)
     
  Misc. Issues  
Ways to Choose r from n-- with repetition -order does not matter n+r-1Cr
Ways to Choose r from n-- without repetition-order does not matter nCr
Ways to Choose r from n- with p things always included n-pCr-p
Ways to Choose r from n- with p things always excluded n-pCr
Ways to Choose r from n-- with repetition -order  matters nr
Ways to Choose r from n-- without repetition-order  matters nPr
Combination of r things out of -1 objects -1Cr = (-1)r
Combination of r things out of -n objects (n > 0) -nCr = (-1)r[n+r-1Cr]