`text(Case 1.)`
Number of ways in which `n` distinct things can be divided into `r` unequal groups containing `a_1, a_2, a_3, ......, a_r` things (different number of things in each group and the groups are unmarked, i.e., not distinct)
`= text()^nC_(a_1) - text()^(n-a_1)C_(a_2) - ... - text()^(n-a_1-a_2 - ... -a_(r-1))C_(a_r)`
`=(n!)/(a_1!a_2!a_3!.....a_r!)`
`text(Case 2.)`
Number of ways in which `n` distinct things can be distributed among `r` persons such that one person get `a_1` things, another person get `a_2` things, ... and another person gets `a_r` things (each person gets different number of things)
`=` Number of ways in which `n` distinct things can be divided into r unequal groups containing `a_1, a_2, a_3, ......, a_r` things (different number of objects in each group and the groups are numbered, i.e., distinct)
`=(n!r!)/(a_1!a_2!a_3!.....a_r!)`
(Note that `a_1+ a_2+ a_3+ ...... +a_r=n`
`text(Case 3.)`
Number of ways in which `m-n` distinct things can be divided equally into `n` groups (each group will have m things and the groups are unmarked, i.e., not distinct)
`=((mn)!)/((m!)^n.n!)`
`text(Case 4.)`
Number of ways in which `m-n` distinct things can be distributed equally among `n` persons (each person gets `m` number of things)
`=` Number of ways in which `m-n` distinct things can be divided equally into `n` groups (each group will have m things and the groups are numbered, i.e., distinct)
`=((mn)!)/((m!)^n)`
`text(Case 1.)`
Number of ways in which `n` distinct things can be divided into `r` unequal groups containing `a_1, a_2, a_3, ......, a_r` things (different number of things in each group and the groups are unmarked, i.e., not distinct)
`= text()^nC_(a_1) - text()^(n-a_1)C_(a_2) - ... - text()^(n-a_1-a_2 - ... -a_(r-1))C_(a_r)`
`=(n!)/(a_1!a_2!a_3!.....a_r!)`
`text(Case 2.)`
Number of ways in which `n` distinct things can be distributed among `r` persons such that one person get `a_1` things, another person get `a_2` things, ... and another person gets `a_r` things (each person gets different number of things)
`=` Number of ways in which `n` distinct things can be divided into r unequal groups containing `a_1, a_2, a_3, ......, a_r` things (different number of objects in each group and the groups are numbered, i.e., distinct)
`=(n!r!)/(a_1!a_2!a_3!.....a_r!)`
(Note that `a_1+ a_2+ a_3+ ...... +a_r=n`
`text(Case 3.)`
Number of ways in which `m-n` distinct things can be divided equally into `n` groups (each group will have m things and the groups are unmarked, i.e., not distinct)
`=((mn)!)/((m!)^n.n!)`
`text(Case 4.)`
Number of ways in which `m-n` distinct things can be distributed equally among `n` persons (each person gets `m` number of things)
`=` Number of ways in which `m-n` distinct things can be divided equally into `n` groups (each group will have m things and the groups are numbered, i.e., distinct)
`=((mn)!)/((m!)^n)`