In how many ways can the following prizes be given away to a class of 20 students : first and second in mathematics; first and second in chemistry; first in physics and first in English?
Given: 20 students.
The number of ways of giving first and second prizes in mathematics to a class of 20 students=20 × 19.
(First prize can be given to any one of the 20 students but the second prize cannot be given to the student that received the first prize so the number of candidates for the second prize is 19.)
The number of ways of giving first and second prizes in chemistry
To a class of 20 students $=20 \times 19$.
The number of ways of giving first prize in physics to a class of 20 students $=20$
The number of ways of giving first prize in English to a class of 20 students $=20$
So total number of ways $=20 \times 19 \times 20 \times 19 \times 20 \times 20=57760000$