in how many different orders can five runners finish a race

in how many different orders can five runners finish a race if no ties are allowed?

Solution

(Problem is with out replacement )

The number of orders that 5 runners can finish a race is 5! or 5X4X3X2X1=120

SInce there are 5 runners there are 5 possibilities for first place.

The possibilities for second are 4 because there are 4 runners left to take second. That makes a total of 20 possibilities for first and second place winners.

Third place will have 3 possible runners after first place and second have finished and fourth and fifth, four and five. That makes a total of 120 possibilities for first,second,third,fourth,and fifth place