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.