Question

A train timetable must be compiled for various days in a 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?
a.) 140
b.) 210
c.) 133
d.) 72

Answer 1

Hint: The above question is a simple question of permutation and combination. We can see from the question that there are 3 different conditions and the total number of days is 7. So, we have to compile a timetable for those 7 days for a week. The number of ways will be given by $ {}^{7}{{C}_{3}}\times {}^{4}{{C}_{2}}\times {}^{2}{{C}_{2}} $ .

Complete step by step answer:
We will solve the above question by utilizing the concept of permutation and combination. We can see from the question that we are given three conditions for which the timetable is to be compiled.
And, the conditions are:
Condition 1: There are two trains which depart for three days
Condition 2: There is one train daily for two days.
Condition 3: There are three trains for two days.
So, we can say that there is a total of seven days and we have compiled the timetable for those 7 days for a week.
And, so for the first condition, the total number of ways of selecting three days out of 7 days a week is $ {}^{7}{{C}_{3}} $ .
 So, we can say that the remaining days are 4. Now, we will select 2 days out of the remaining 4 for the second condition by $ {}^{4}{{C}_{2}} $ ways.
Now, we will, at last, select the remaining 2 days for condition 3 by $ {}^{2}{{C}_{2}} $ ways.
So, the total number of ways in which the timetable can be compiled will be equal to $ {}^{7}{{C}_{3}}\times {}^{4}{{C}_{2}}\times {}^{2}{{C}_{2}} $ because we will select the number of days simultaneously.
So, number different timetables can be compiled that can be compiled = $ {}^{7}{{C}_{3}}\times {}^{4}{{C}_{2}}\times {}^{2}{{C}_{2}} $
 $ =\dfrac{7!}{3!\times 4!}\times \dfrac{4!}{2!\times 2!}\times 1 $
  $ =\dfrac{7\times 6\times 5}{3\times 2\times 1}\times \dfrac{4\times 3}{2} $
 $ =210 $
This is our required solution.
Hence, option (b) is our correct answer.

Note:
 We can also solve the above question by another method. We can see that we have the same condition 1 in the timetable is for 3 days, condition 2 for 2 days, and condition 3 for also 2 days. So, number of different timetables that can be compiled = $ \dfrac{7!}{3!\times 2!\times 2!} $ = 210.