If a family has 7 children, in how many ways could the parents have 5 boys and 2 girls?

Question

If a family has 7 children, in how many ways could the parents have 5 boys and 2 girls?

Answer 1

If all the children were you unique, you count the number of orderings using the factorial (!), which is the product of an integer and all positive integers smaller than it. (e.g. 5! = 5*4*3*2*1 = 120)

 

To find the orderings of a group with subgroups of identical items (i.e. all boys and all girls), you divide the total number of orderings for the overall group (7!) by the number of orderings of each subgroup (5!, 2!).

7! / ( 5! 2! )

 

A neat trick with dividing factorials is that you can write the larger factorial using smaller factorials. In this case 7! = 7*6*5!

 

(7*6*5!) / ( 5! 2! )

 

The 5!s cancel, and you get 7*6 / 2 = 21.