Birthday problem with repetitions

[wintermute++]

Homework Statement

How many different sets of birthdays are available with k people and 365 days when we don’t distinguish the same
birthdays in different orders?

Homework Equations

[/B]
I approached this using what was proven in a previous problem, provided I did that right. This what I had:

$C_{n+k-1, k}$

for combinations that involve repetitions.

The Attempt at a Solution

$C_{365+k-1,k}$

Not much of an attempt at the solution since it seemed obvious enough. But the book says the answer is $C_{365+k,k}$ and I'm struggling to get to this solution.

 

[haruspex]

I agree with your answer. As a check, we can consider a much simplified version: 2 days, 2 people. C_2+2-1,2 = 3: (1,1), (1,2), (2,2).

 

[wintermute++]

Thanks haruspex. You've been a great help for me so far.