There are 5 cyclists in a race. There will be a first-place and a second-place prize awarded. In how many different ways can the 2 prizes be awarded?

Question

 There are
5
cyclists in a race. There will be a first-place and a second-place prize awarded. In how many different ways can the
2
prizes be awarded?

Answer 1

Call the cyclists A, B, C, D, and E.

 

How many possibilities for first place?  5    They are A, B, C, D, and E.

 

Once a first place is awarded, how many possibilities for second place?  4 for each first place awarded

  They are:

     AB, AC, AD, AE

     BA, BC, BD, BE

     CA, CB, CD, CE

     DA, DB, DC, DE

     EA, EB, EC, ED

 

That is 5*4 = 20 possibilities.

 

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement.

 

       nPr = n! / (n - r)! 

       ₅P₂ = 5! / (5-2)!

       ₅P₂ = (5*4*3*2*1) / (3*2*1) = 5*4 = 20