Would I use combinations or permutations?If 6 people are running in a race, how many possible ways can they come in if there are no ties and...

Permutation is as good as arranging. Whereas,in selecting a group of persons, we are interested only in number of persons irrespective of the order of selection. So we use combinations .

Let A,B,C,D,E,F be the 6 persons in race.

A may come in any of the 6 places. So, 6 choice.

For any one place of A, B can occupy any one of the other 5 places.

For any particular order of A and B, C ha the choice of the places  6-2=4 places.

Similarly D has the choice of 6-3= 3places.

E can occupy 2 places and F has to be at the available one place.

Thus, the choces being free and indepedent, the number of aarrangements or ways is 6P6 given by:

6P6 = 6(6-1)(6-2)(6-3)(6-4)(6-5) = 6! = 720