How many groups can be made with 12 boys and 16 girls if the same number of girls and boys are in each group?

Answer 1

You haven't said what's the total size of the group.

Here are all of them:

Groups of 2: (12) x (16) = 192

Groups of 4: (12 x 11) x (16 x 15) / (2!)2 = 7,920

Groups of 6: (12 x 11 x 10) x (16 x 15 x 14) / (3!)2 = 123,200

Groups of 8: (12 x 11 x 10 x 9) x (16 x 15 x 14 x 13) / (4!)2 = 900,900

Groups of 10: (12 x 11 x 10 x 9 x 8) x (16 x 15 x 14 x 13 x 12) / (5!)2 = 3,459,456

Groups 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 = 7,399,392

Groups 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 = 9,060,480

Groups 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 = 6,370,650

Groups 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 = 2,516,800

Groups 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 = 528,528

Groups 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 = 52,416

Groups 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 = 1,820

You 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 = 30,421,754

You're welcome.