JEE Main & Advanced Mathematics Permutations and Combinations Division Into Groups

Case I : (1) The number of ways in which \[n\] different things can be arranged into \[r\] different groups is \[^{n+r-1}{{P}_{n}}\] or n ! \[^{n-1}{{C}_{r-1}}\] according as blank group are or are not admissible.

(2) The number of ways in which  \[n\] different things can be distributed into \[r\] different group is

\[{{r}^{n}}{{-}^{r}}{{C}_{1}}{{(r-1)}^{n}}{{+}^{r}}{{C}_{2}}{{(r-2)}^{n}}-.........+{{(-1)}^{n-1}}{{\,}^{n}}{{C}_{r-1}}\] or Coefficient of \[{{x}^{n}}\] is n ! \[{{({{e}^{x}}-1)}^{r}}\].

Here blank groups are not allowed.

(3) Number of ways in which \[m\times n\] different objects can be distributed equally among \[n\] persons (or numbered groups) = (number of ways of dividing into groups) \[\times \] (number of groups)\[!=\frac{(mn)\,!\,n\,!}{{{(m\,!)}^{n}}n!}=\frac{(mn)\,!}{{{(m!)}^{n}}}\].

Case II : (1) The number of ways in which \[(m+n)\] different things can be divided into two groups which contain \[m\] and \[n\] things respectively is, \[^{m+n}{{C}_{m}}{{.}^{n}}{{C}_{n}}=\frac{(m+n)\,!}{m\,!\,n!},m\ne n\].

Corollary: If \[m=n\], then the groups are equal size. Division of these groups can be given by two types.

Type I : If order of group is not important : The number of ways in which \[2n\] different things can be divided equally into two groups is \[\frac{(2n)!}{2!{{(n!)}^{2}}}\].

Type II : If order of group is important : The number of ways in which 2n  different things can be divided equally into two distinct groups is \[\frac{(2n)!}{2!{{(n!)}^{2}}}\times 2!=\frac{2n!}{{{(n!)}^{2}}}\].

(2) The number of ways in which \[(m+n+p)\] different things can be divided into three groups which contain m, n and p things respectively is

\[^{m+n+p}{{C}_{m}}{{.}^{n+p}}{{C}_{n}}{{.}^{p}}{{C}_{p}}=\frac{(m+n+p)\,!}{m\,!n\,!\,p\,!},\,m\ne n\ne p\].

Corollary : If \[m=n=p\], then the groups are equal size. Division of these groups can be given by two types.

Type I : If order of group is not important : The number of ways in which 3p different things can be divided equally into three groups is \[\frac{(3p)\,!}{3!{{(p!)}^{3}}}\].

Type II : If order of group is important : The number of ways in which \[3p\] different things can be divided equally into three distinct groups is \[\frac{(3p)!}{3!{{(p!)}^{3}}}3!=\frac{(3p)\,!}{{{(p!)}^{3}}}\].

(i) If order of group is not important : The number of ways in which \[mn\] different things can be divided equally into m groups is \[\frac{mn!}{{{(n!)}^{m}}m!}\].

(ii) If order of group is important: The number of ways in which \[mn\] different things can be divided equally into \[m\] distinct groups is \[\frac{(mn)!}{{{(n!)}^{m}}m!}\times m!=\frac{(mn)!}{{{(n!)}^{m}}}\].