Answer
Yes
Work Step by Step
Yes. This follows from the generalized pigeonhole principle with 2000 pigeons, 366 pigeonholes, and k = 5, using the fact that 2000 > 5*366. Alternatively, since 5 < 2000/366 (k < n/m), then there is at least 5+1=6 people occupy the same birthday. Therefore, there must be at least 5 people with the same birthday (since 6 people > 5 people).
2000 people/366 days (including 1 extra day during leap year) = 5.46
2000 pigeons, 366 pigeonholes, k = 5
2000 > (5 * 366 = 1830)
