n biscuits are distributed among N beggars at random. The probability that a particular beggar gets r( <n) biscuits, is

${^nC}_r \left(\frac{1}{N}\right)^r \left(\frac{N-1}{N}\right)^{n-r}$

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 distributed n biscuits among N beggars is

$N × N × .....× N = N^n$

n - times

Now, r biscuits can be given to a particular beggar in ${^nC_r}$, 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

${^nC_r} ×  (N-1)^{n-r}$

Hence, required probability $=\frac{^nC_r ×(N-1)^{n-r}}{N^n}$