Question

Suppose you teach a class of twelve students that you wish to assign to three groups of four students each. In how many different ways can groups be made?

Answer 1

Solution :

The first group can be chosen from the 12 people in ₁₂ C ₂ ways, the second group can be chosen from

the remaining 8 people in ₈ C ₄ ways, and the third group is formed with the remaining people .

( ₁₂ C ₄ * ₈ C _{4 * 4} C ₄ ) / 3!

_n C _x = n! / x!(n - x)!

n! = n(n - 1)(n - 2) .... and so on .

n = 12 and x = 4

₁₂ C 4 = 12! / 4!(12 - 4)! = 12! / 4!(8)! = 12 * 11 * 10 * 9 / 4 * 3 * 2 = 495

₈ C ₄ = 8! / 4! (8 - 4)!! = 8! / 4!* 4! = 70

₄ C ₄ = 1

3! = 3 * 2 = 6

( ₁₂ C ₄ * ₈ C _{4 * 4} C ₄ ) / 3! = (495 * 70 * 1) / 6 = 34650 / 6 = 5775

5775  different ways can groups be made .