setting up by use generating functions to answer the following question: How many ways can you distribute identical pieces of candy to 5 children (Alice, Bob, Charlie, Dan, and Ellen) such that all of of the following conditions are satisfied (simultaneously): i. Twenty pieces of candy are distributed in total. ii. Alice gets at least 2 pieces of candy. iii. Bob gets at most 3 pieces of candy. iv. The number of pieces of candy that Charlie gets is a multiple of 3 (zero is allowed). v. Dan gets 2, 6, or 9 pieces of candy. vi. Ellen can get any number of pieces of candy Solution it is 40 number of ways they can distribute first 20 pieces are distrubuted so in that alice gets 2 thaats comon and bob gets max of three ie 0 to 3 and charlie gets multiples of 3 ie 0,3,6,9 max and dan gets onlt 2 , 6 ,9 so alice if fixeed so there would be 18 left and so by doing permutaon we get it.

1. Seven (distinct) car accidents occurred in a week. What is the probability that they occurred on
the same day?
Solution
Given X~Poisson(=7/7=1 per day)
P(X=x)=(exp(-1))/x!
SO the probability is
P(X=7)=(exp(-1))/(1*2*3*4*5*6*7)=0.00007