Question

1. There are 43 cars that start in every Nascar Sprint Cup race. How many ways can they place in the top 5 at each race?

Answer 1

Question 1)

Total Number of Cars = 43
Number of positions = 5

We have to arrange the cars for first 5 positions. This is a problem of permutations and is equivalent to form arrangements of 5 objects from 43 objects. 

So, the number of ways to place the top 5= 43P5

$= \frac{43!}{(43-5)!} \\ \\ = \frac{43!}{38!} \\ \\ =115511760$

Thus, there are 115511760 ways to place the 43 cars in top 5.

Question 2)

Total number of students = 102
Number of students that should be together = 2

Making pairs from a group of students is equivalent to making combinations. So this is a problem of combinations and is equivalent to forming combination of 2 objects from 102.

So, the number of pairs that can be formed = 102C2
$= \frac{102!}{2!*100!} \\ \\ =5151$

This mean with 102 students 5151 pairs of students are possible.