How many cars are in a race if there are 156 ways for 2 cars to finish first and second?

Answer 1

13 cars.

Let there be n cars in the race.

Then the number of different two cars first and second is n(n-1) = 156.

There are two ways forward: either:

solve the quadratic: n(n-1) - 156 = 0

→ n² - n - 156 = 0

→ n = (1 ± √(1² - 4×-156)/2

→ n = (1 ± √265)/2

→ n = (1 ± 25)/2

→ n = 13 or -12

Can't have -12 cars, so there must be 13

or:

Realise we need a factor pair of 156 that has a difference of 1. The factor pairs of 156 are:

1 x 156

2 x 78

3 x 52

4 x 39

6 x 26

12 x 13

with the last pair 12 x 13 with the required difference of 1. n is the larger of the two, so the number of cars is 13.

Answer 2

There is no possible answer. The solution to the relevant equation is 18.17.. and it is not possible to have a fraction of a car in a race.