Question:
Ten students are participating in a race. In how many ways can the first three prizes be won?
Solution:
To find: number of ways of winning the first three prizes
The first price can go to any of the 10 students.
The second price can go to any of the remaining 9 students.
The third price can go to any of the remaining 8 students.
Formula:
Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is
$P(n, r)=n ! /(n-r) !$
Therefore, a permutation of 10 different objects in 3 places is
$P(10,3)=\frac{10 !}{(10-3) !}$
$=\frac{10 !}{7 !}=\frac{3628800}{5040}=720$
Therefore, there are 10 × 9 × 8 = 720 ways to win first three prizes.