Question

(80 pts) How many different ways can a captain and a co-captain be selected for a rugby team if there are 10 players?

Answer 1

45 if two co-captains or 90 if captain and co-captain are two DISTINCT roles.

Explanation:

METHOD ONE

Label each player (either a through j OR 1 - 10).

List combinations:

1. a b
2. a c
3. a d
4. a e
5. a f
6. a g
7. a h
8. a i
9. a j
10. b c (not "b a" because that would be the same as #1)
11. b d
12. b e
13. b f
14. b g
15. b h
16. b i
17. b j
18. c d (not "c a" or "c b" because those would be duplicates)
19. c e
20. c f
21. c g
22. c h
23. c i
24. c j
25. d e
26. d f
27. d g
28. d h
29. d i
30. d j
31. e f
32. e g
33. e h
34. e i
35. e j
36. f g
37. f h
38. f i
39. f j
40. g h
41. g i
42. g j
43. h i
44. h j
45. j i

METHOD 2

use formula

Answer 2

There are 90 different ways

Explanation:

We can apply factorials to help solve this equation.

Just to review, 4!= 4x3x2x1

To solve this problem, we need to use factorials but only a portion of it.

Since there are only 2 roles, we only have to go down two numbers

For captain, there are 10 players available,

But for co captain, there are 9, because the captain can't also be the co captain

So the result is 10x9

We don't have to go further since there are no more further roles to occupy

10x9=90

There are 90 different ways