There is actually a lot more than just 3 birds of Paradise. Here is a full list:

1. Paradise-crow
2. Glossy-mantled Manucode
3. Jobi Manucode
4. Crinkle-collared Manucode
5. Curl-crested Manucode
6. Trumpet Manucode
7. Long-tailed Paradigalla
8. Short-tailed Paradigalla
9. Arfak Astrapia
10. Splendid Astrapia
11. Ribbon-tailed Astrapia
12. Princess Stephanie's Astrapia
13. Huon Astrapia
14. Arfak Parotia
15. Queen Carola's Parotia
16. Lawes's Parotia
17. Eastern Parotia
18. Wahnes's Parotia
19. King of Saxony Bird-of-paradise
20. Superb Bird-of-paradise
21. Magnificent Riflebird
22. Growling Riflebird
23. Paradise Riflebird
24. Victoria's Riflebird
25. Black Sicklebill
26. Brown Sicklebill
27. Black-billed Sicklebill
28. Pale-billed Sicklebill
29. Magnificent Bird-of-paradise
30. Wilson's Bird-of-paradise
31. King Bird-of-paradise
32. Standardwing
33. Twelve-wired Bird-of-paradise
34. Lesser Bird-of-paradise
35. Greater Bird-of-paradise
36. Raggiana Bird-of-paradise
37. Goldie's Bird-of-paradise
38. Red Bird-of-paradise
39. Emperor Bird-of-paradise
40. Blue Bird-of-paradise

As you can see there are 40 different species of bird of paradise which is a lot more than some people think. The 3 most famous are the king, Magnificent and Raggiana bird-of-paradise.