If `n` biscuits are distributed among `N` beggars, find the chance that a particular beggar will get `r(

Options

A. `((N-1)^(n-r))/(N^(n))`
B. `(.^(n)C_(r ))/(N^(n-r))`
C. `(.^(n)C_(r )(N-1)^(r ))/(N^(n))`
D. `(.^(n)C_(r )(N-1)^(n-r))/(N^(n))`

Answer

D

Explanation

Since a biscuit can be given to any one of N beggars. Therefore, each biscuit can be distributed in N ways. So, the total number of ways of distributing n biscuits among N beggars is
`N xx N xx ..xx N=N^(n)`
n-times
Now, r biscuits can be given to a particular beggar in `.^(n)C_(r )` ways and the remaining (n-r) biscuits can be distributed to (N-1) beggars in `(N-1)^(n-r)` ways. Thus, the number of ways in which a particular beggar receives r biscuits is
`.^(n)C_(r )xx(N-1)^(n-r)`
Hence, required probability `=(.^(n)C_(r )xx(N-1)^(n-r))/(N^(n))`