A train timetable must be compiled for various days of the week so that two trains twice a day depart for three days,one train daily for two days and three trains once a day for two days.How many different timetables can be compiled?
I could not understand this problem,its answer is given to be pretty simple $\frac{7!}{2!2!3!}$.I did not understand how they arrived at this answer.
The problem is quite badly formulated; I'm not surprised that you didn't understand it.
Judging from the answer, it seems that they mean that there are three types of days which certain requirements for the train timetable, the details of which are irrelevant; there should be three days of the first type, two days of the second type and two days of the third type.
This kind of problem is solved by the multinomial coefficients that count the number of ways of classifying a set of objects into a specified number of types with specified numbers of occurrences for the types. In the present case, the multinomial coefficient is
$$ \binom7{2,2,3}=\frac{7!}{2!2!3!}\;. $$
Imagine distributing the $7$ days onto three bins with the specified capacities: $AAA\mid BB\mid CC$. There are $7!$ ways to do this, but the order within each category doesn't matter, so we need to compensate for the overcounting by dividing by the $3!$ irrelevant orders within category $A$ and likewise by $2!$ and $2!$ for $B$ and $C$.