GATE CS 1999 Q2.2 — Combinatorics: Counting, Recurrence Relations, Generating Functions

To find the number of ways the two girls (say, $G_1$ and $G_2$) can divide the flowers among themselves, we assume that flowers of the same type are identical.

For each type of flower, the number of ways to distribute $n$ identical items between $2$ people is equivalent to finding the number of non-negative integer solutions to $x_1 + x_2 = n$, which is simply $n + 1$ ways (since $G_1$ can receive any number of flowers from $0$ to $n$, uniquely determining $G_2$'s share).

• Roses ($10$ identical):
  Number of ways to distribute = $10 + 1 = 11$ ways.
• Sunflowers ($15$ identical):
  Number of ways to distribute = $15 + 1 = 16$ ways.
• Daffodils ($15$ identical):
  Number of ways to distribute = $15 + 1 = 16$ ways.

Since the distribution of each type of flower is independent, we use the product rule to find the total number of ways:

Since $2816$ is not listed in options A, B, or C, the correct choice is D.