Calculus 3 Combinations and Permutations

Question

On a business retreat, your company of 24 executives go golfing.

a. You need to divide up into foursomes (groups of four people): a first foursome, a second foursome, and so on. How many ways can you do this.

b. After all your hard work, your realize that in fact, you want each foursome to include one of the six Board members. How many ways can you do this

How many different seating arrangements are possible for King Arthur and 5 of his knights around their table?

Suppose there was a throne at the table and King Arthur must sit there. How many seating arragements are possible in this case?

Answer 1

Suppose that the order of foursomes is important: what people are in the first foursome, what are in the second foursome and so on.

a. (₂₄C₄) (₂₀C₄) (₁₆C₄) (₁₂C₄) (₈C₄) (₄C₄) = 24! / (4!)⁶ = 24! / 24⁶ = 23! / 24⁵

^b. (6*₁₈C₃) (5*₁₅C₃) (4*₁₂C₃) (3*₉C₃) (2*₆C₃) (1*₃C₃) = 6! * 18! / (3!)⁶ = 17! *5 / 18

If the order of foursomes is not important, these answers must be divided by 6!.

Answer 2

a. ₂₄C₄ = ?

b. 6 * ₁₈C₃ = ?

Draw Arthur's table.