Question

3. Let an be the number of ways to select n marbles from a large supply of blue, red, and yellow marbles where the selection must include an even number of blue ones. Write down a generating function for the sequence an in closed form.

Answer 1

3. Let an be the number of ways to select n marbles from a large supply of blue, red, and yellow marbles where the selection

3. Let an be the number of ways to select a marbles from a large supply of blue, red. and yellow marbles where the selection must include an even number of blue ones, write down a generating function for the sequence an in closed form. Ans: A generating function is a way of encoding an infinite Sequence of numbers (and by treating them as the coefficients of a power Series. This formal power Series in the generating function. We need to find the number of ways firstly: by which an can select "n' marbles from a large Supply of blue, red and yellow marbles Call of the Same Size) if the selection must include an even number of blue ones. for the selection in which we includes an even number of blue ones, the generating function

Since the number of f blue marbles must be even, there can be o, 2,4,6 and so on blue marbles. Similarly, the number of red & yellow marbles can be 0,1,2,3 and so on. So for the other two Selections, where selections is restricted, the generating function is (1+x+3x2+.....) So, the generating finction for the given Sletion is f(x) = (1+x2244 --) (1+2+327..?