Question

Find the generating function to determine the number of ways to pick k objects from n objects when the ith object can appear
i+jn
times for and any integer $j\geq 0$ .

Answer 1

The generating function to ensure that the ithith object appears at least n+in+i times is as follows:

g(x) = (x1+x2+…+xk ) (x1+n+x2+n…+xk ) (x1+2n+x2+2n…+xk)... (xn*n+1+xn*n+2…+xk )

Here, the power of x in the first term of the product represents the number of times the first object is picked. Since the first object appears 1+(0)n = 1 time, the smallest power of x in the first term is 1. The maximum number of objects to be chosen is k, and hence, the maximum power is k. So, the number of ways to choose k objects is the coefficient of xk in the generator function g(x).