Question

On planet K, there is 4 political groups K major, K minor, K left, K right. The High Council voted on a bill that some wanted to make a law. Here is a table that represents the vote.

	K major	K minor	K left	K right	Total
Yes	14	12	6	5	37
No	5	10	14	8	37
Abstained	7	2	7	16	32
Total	26	24	27	29	106

a) How many people are on the High Council

b) If 10 council members are picked at random (Statement for questions 1b-5b)

1b) How many ways can you pick any 10 council members

2b) How many ways can you get all of them to be K Major council members

3b) How many ways can you get 3 to be in K Minor, and 3 to be in K Right council member

4b) How many ways can you get 6 who voted no

5b) How many ways can you get 3 who were both K Minor and voted Yes

Answer 1

a) There are total 106 people in the council       
        
1b) 10 council members can be picked from 106 people in 106C10 ways       
   = 31853506369685     
        
2b) There are total 26 K-Major council members       
Selecting 10 council members from 26 K-Majors can be done in 26C10 ways       
   = 5311735 ways      
        
3b) Total K-Minors = 24  , Total K-Rights = 29     
3 K-Minors can be selected in 24C3 ways = 2024 ways       
3 K-Right can be selected in 29C3 ways = 3654 ways       
Remaining 4 members have to be selected from (106-24-29 = 53) members       
These can be selected in 53C4 ways = 292825       
Total number of ways = 2024 * 3654 * 292825       
    = 2165644681200 ways     
        
4b) Total number of members who voted No = 37       
6 members who voted NO can be selected in 37C6 ways = 2324784 ways       
Remaining 4 members have to be selected from (106-37 = 63) members       
These can be selected in 63C4 ways = 595665       
Total number of ways = 2324784 * 595665       
    = 1384792461360 ways      
        
5b) Number of members who are both K-Minor and voted YES = 12       
3 such members can be selected in 12C3 ways = 220 ways       
Remaining 7 members have to be selected from (106-12 = 94) members       
These can be selected in 94C7 ways = 10235867928       
Total number of ways = 10235867928 * 220       
    = 2251890944160 ways