From a candidate pool of 22 people (12 men and 10 women), a jury of 12 people is selected.

Question

how many different 12-person juries with the same number of men and women can be selected from this jury pool?

how many different 12-person juries can be selected from this jury pool?

once the jury of 12 people is selected, from these 12 we must select people to fill the offices of chairperson, vice-chairperson, and recording secretary. In how many different ways can the offices be filled from these twelve people?

Answer 1

An equal number of men and women means there must be 6 of each. The number of ways of selecting a combination of 6 out of 12 men is 12*11*10*9*8*7/6*5*4*3*2*1=924; the number of ways of selecting a combination of 6 out of 10 women is 210; therefore there 210*924=194,040 ways of selecting the jury.

No more than 10 women can be selected, so if all the women are selected then 2 men out of 12 are to be selected=66 ways.

If 9 women out of 10 are selected then the number of ways is 10; for selecting 3 out of 12 men=220. Combine these figures=2,200.

Then 8/10 women * 4/12 men=22,275; 7/10 women * 5/12 men=95,040; 6/10*6/12=194,040; 5/10*7/12=199,584; 4/10*8/12=103,950; 3/10*9/12=26,400; 2/10*10/12=2,970; 1/10*11/2=120; all men=1.

Total ways: 66+2200+22275+95040+194040+199584+103950+26400+2970+120+1=646,646.

There are 12 choices for chair, leaving 11 for vice-chair, then 10 for secretary=12*11*10=1,320 ways of picking 3 officers times 6 ways of arranging the officers to fit the posts: 6*1320=7,920.