Answer to Question #320505 in Statistics and Probability for Klum

Question

There are N urns in which M balls are randomly scattered. Find the probability p that the given (for example, the first) urn contains exactly k balls.

Answer 1

$The\,\,number\,\,of\,\,ways\,\,to\,\,scatter\,\,M\,\,balls\,\,in\,\,N\,\,urns\,\,is\,\,N^M,\\\sin ce\,\,for\,\,each\,\,ball\,\,there\,\,are\,\,N\,\,posibilities.\\The\,\,number\,\,of\,\,ways\,\,to\,\,scatter\,\,balls\,\,so\,\,tht\,\,the\,\,first\,\,urn\,\,contains\,\,k\,\,balls\\is\,\,the\,\,number\,\,of\,\,ways\,\,to\,\,select\,\,k\,\,balls\,\,to\,\,the\,\,first\,\,urn\,\,\\multiplied\,\,by\,\,the\,\,number\,\,of\,\,ways\,\,to\,\,scatter\,\,other\,\,M-K\,\,balls\,\,to\,\,N-1 urns,\\that\,\,is\,\,C_{M}^{K}\cdot \left( N-1 \right) ^{M-K}\\The\,\,probability\,\,is\\\frac{C_{M}^{K}\left( N-1 \right) ^{M-K}}{N^M}$