Distinct arrangement of n number of objects among r distinct cells (without ordering inside a cell)
* 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-1}Cn | |
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-1}Cr-1 | |
Each cell to have at least q objects | ^{n+r-1-rq}Cr-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) | r^{n} | |
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-1}Cr | |
Ways to Choose r from n-- without repetition-order does not matter | ^{n}Cr | |
Ways to Choose r from n- with p things always included | ^{n-p}Cr-p | |
Ways to Choose r from n- with p things always excluded | ^{n-p}Cr | |
Ways to Choose r from n-- with repetition -order matters | n^{r} | |
Ways to Choose r from n-- without repetition-order matters | ^{n}Pr | |
Combination of r things out of -1 objects | ^{-1}Cr = (-1)^{r} | |
Combination of r things out of -n objects (n > 0) | ^{-n}Cr = (-1)^{r[n+r-1}Cr] | |