`(a)` `text(Division of Objects Into Groups of Unequal Size)`
Theorem Number of ways in which (m + n) distinct objects ean be divided into two unequal groups containing m and n objects is
`((m+n)!)/(m!n!)`
`text(Corollary I)` The number of ways to distribute (m + n) distinct objects among 2 persons in the groups containing m and n objects
=(Number of ways to divide) x (Number of groups)
`= ((m+n))/(m!n!) xx 2!`
`text(Corollary II)` The number of ways in which (m + n + p) distinct objects can be divided into three unequal groups containing m, n and p objects, is
`text( )^(m+n+p)C_m. text( )^(n+p)C_n. text( )^pC_p = ((m+n+p)!)/(m!n!p!)`
`text(Corollary III)` The number of ways to distribute (m + n + p) distinct objects among 3 persons in the groups containing m, nand p objects = (Number of ways to divide) x (Number of groups)
`= ((m+n+p)!)/(m!n!p!)`
`text(Corollary IV)` The number of ways in which `(x_1 + x_2 + x_3 + ... + x_n )` distinct objects can be divided into n unequal groups containing `x_1, x_2, ... , X_n` objects, is
`((x_1+x_2+x_3 + ........ + x_n)!)/(x_1! x_2!x_3!...........x_n!)`
`text(Corollary V)` The number of ways to distribute `(x_1 + x_2 + x_3 + ... + x_n )` distinct objects among n persons in the groups containing `x_1 , x_2 , ......... , x_n` objects
= (Number of ways to divide) x (Number of groups)
`=((x_1+x_2+x_3 + ........ + x_n)!)/(x_1! x_2!x_3!...........x_n!) xx n!`
`(b)` `text(Division of Objects Into Groups of Equal Size)`
The number of ways in which mn distinct objects can be divided equally into m groups, each containing n objects and
(i) If order of groups is not important is
`= ((mn)!)/((n!)^m) xx 1/(m!)`
(ii) If order of groups is important is
`(((mn)!)/(n!)^m xx 1/(m!)) xx m = ((mn)!)/((n!)^m)`
Note Division of 14n objects into 6 groups of 2n, 2n, 2n, 2n, 3n, 3n, size is
`(((14n)!)/((2n)!(2n)!(2n)!(2n)!(3n)!(3n)!))/(4!2!) = (14n)/(((2n)!)^4((3n)!)^2) xx 15`
`(a)` `text(Division of Objects Into Groups of Unequal Size)`
Theorem Number of ways in which (m + n) distinct objects ean be divided into two unequal groups containing m and n objects is
`((m+n)!)/(m!n!)`
`text(Corollary I)` The number of ways to distribute (m + n) distinct objects among 2 persons in the groups containing m and n objects
=(Number of ways to divide) x (Number of groups)
`= ((m+n))/(m!n!) xx 2!`
`text(Corollary II)` The number of ways in which (m + n + p) distinct objects can be divided into three unequal groups containing m, n and p objects, is
`text( )^(m+n+p)C_m. text( )^(n+p)C_n. text( )^pC_p = ((m+n+p)!)/(m!n!p!)`
`text(Corollary III)` The number of ways to distribute (m + n + p) distinct objects among 3 persons in the groups containing m, nand p objects = (Number of ways to divide) x (Number of groups)
`= ((m+n+p)!)/(m!n!p!)`
`text(Corollary IV)` The number of ways in which `(x_1 + x_2 + x_3 + ... + x_n )` distinct objects can be divided into n unequal groups containing `x_1, x_2, ... , X_n` objects, is
`((x_1+x_2+x_3 + ........ + x_n)!)/(x_1! x_2!x_3!...........x_n!)`
`text(Corollary V)` The number of ways to distribute `(x_1 + x_2 + x_3 + ... + x_n )` distinct objects among n persons in the groups containing `x_1 , x_2 , ......... , x_n` objects
= (Number of ways to divide) x (Number of groups)
`=((x_1+x_2+x_3 + ........ + x_n)!)/(x_1! x_2!x_3!...........x_n!) xx n!`
`(b)` `text(Division of Objects Into Groups of Equal Size)`
The number of ways in which mn distinct objects can be divided equally into m groups, each containing n objects and
(i) If order of groups is not important is
`= ((mn)!)/((n!)^m) xx 1/(m!)`
(ii) If order of groups is important is
`(((mn)!)/(n!)^m xx 1/(m!)) xx m = ((mn)!)/((n!)^m)`
Note Division of 14n objects into 6 groups of 2n, 2n, 2n, 2n, 3n, 3n, size is
`(((14n)!)/((2n)!(2n)!(2n)!(2n)!(3n)!(3n)!))/(4!2!) = (14n)/(((2n)!)^4((3n)!)^2) xx 15`