(1) How many different teams can be selected? (2) How many different teams can be selected if each team must contain exactly 4 girls and 3 boys? (3) How ma

Question

Assume that there are 9 girls and 10 boys in the neighborhood club, and a team of 7 is to be selected.

Answer 1

Question 1. Assume that there are 9 girls and 10 boys in the neighborhood club, and a team of 7 is to be selected. How many different teams can be selected?

Step 1. Permutation or combination.

Ask yourself: Does the order of choosing team members matter? Does the question specify any roles they will take?

No, so the order does not matter, and it is a combination problem.

Step 2. Apply Equation

NCn = N! / n!(N-n)! (Read: Big N (19) choose little n (7))

19C7= 19*18*17*16*15*14*13*12! / 7!(12!)

Note the 12! Divided by another 12!, they cancel out.

19*18*17*16*15*14*13 / 7! = 50,388 different combinations of teams by choosing 7 from 19. 

Question 2. How many different teams can be selected if each team must contain exactly 4 girls and 3 boys?

Step 1. Calculate the combinations of teams that have exactly 4 girls from a total of 9 girls (note, this is still a combination question)

NCn = N! / n!(N-n)! (Read: Big N (9) choose little n (4))

9C4= 9*8*7*6*5 / 4!(9-4)!

9C4= 9*8*7*6*5! / 4!*5! (Note 5! Cancel out.)

9C4= 9*8*7*6 / 4! = 126 Combinations of a four-girl team chosen from 9.

Step 2. Calculate the combinations of teams that have exactly 3 boys from a total of 10

NCn = N! / n!(N-n)! (Read: Big N (10) choose little n (3))

10C3 = 10! / 3!(10-3)!

10C3 = 10*9*8*7! / 3!*7! (Note 7! Cancel out)

10C3 = 10*9*8* / 3! = 120 combinations of a three-boy team chosen from 10.

Step 3. Multiply the total combinations of three boy and 4 girl teams.

Combinations of teams with exactly 4 girls and 3 boys: 126*120= 15,120 combinations.