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

Question

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?

Answer 1

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Answer 2

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.