Distribution
Table of Contents
1. Distribution of identical objects
- No. of ways in which \( n \) identical objects can be distributed to \( r \) persons, or number of non-negative integral solutions of the equation \( x_1 + x_2 + \ldots x_r = n \) is
\[ ^{n+r-1}C_{r-1} \] - No. of ways in which \( n \) identical objects can be distributed to \( r \) persons such that each get atleast one object, or number of positive integral solutions of the equation \( x_1 + x_2 + \ldots x_r = n \) is
\[ ^{n-1}C_{r-1} \] - No. of dearrangements:
\[D_n = n! \left( 1 - \frac{1}{1!} + \frac{1}{2!} \ldots \frac{(-1)^{n}}{n!} \right) \]
Recursive relation in dearrangements:
\[ D_n = (n-1) (D_{n-1} + D_{n-2}) \]