How many groups can be made with 12 boys and 16 girls if the same number of girls and boys are in each group?
You haven't said what's the total size of the group.Here are all of them:Groups of 2: (12) x (16) = 192Groups of 4: (12 x 11) x (16 x 15) / (2!)2 = 7,920Groups of 6: (12 x 11 x 10) x (16 x 15 x 14) / (3!)2 = 123,200Groups of 8: (12 x 11 x 10 x 9) x (16 x 15 x 14 x 13) / (4!)2 = 900,900Groups of 10: (12 x 11 x 10 x 9 x 8) x (16 x 15 x 14 x 13 x 12) / (5!)2 = 3,459,456Groups of 12: (12 x 11 x 10 x 9 x 8 x 7) x (16 x 15 x 14 x 13 x 12 x 11) / (6!)2 = 7,399,392Groups of 14:(12 x 11 x 10 x 9 x 8 x 7 x 6) x (16 x 15 x 14 x 13 x 12 x 11 x 10) / (7!)2 = 9,060,480Groups of 16:(12 x 11 x 10 x 9 x 8 x 7 x 6 x 5) x (16 x 15 x 14 x 13 x 12 x 11 x 10 x 9) / (8!)2 = 6,370,650Groups of 18:(12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4) x (16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8) / (9!)2 = 2,516,800Groups of 20:(12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3) x (16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7) / (10!)2 = 528,528Groups of 22:(12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2) x(16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6) / (11!)2 = 52,416Groups of 24:(12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) x(16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5) / (12!)2 = 1,820You can't make groups of more than 24, because there are only 12 boys,and the group must have the same numbers of boys and girls.Total number of possible groups of all sizes = 30,421,754You're welcome.