Question

The ways in which the time table for Monday be completed if there must be 5 lessons that day (algebra, geometry, calculus, trigonometry, vectors) and algebra and geometry must not immediately follow each other are:
Option A: 72
Option B: $5!$
Option C: $\dfrac{{35!}}{2}$
Option D: $6!$

Answer 1

To solve this question, instead of going in a direct way, we can find all the possible ways of studying all the 5 lessons without any condition. Then find the total number of ways in which one can study the 5 lessons such that algebra and geometry are always immediate in the schedule. We can subtract the two values to find the number of ways one can study 5 lessons such that geometry and algebra are never immediate.

Complete step-by-step answer:
Let’s list all the 5 lessons: Algebra, Geometry, Calculus, Trigonometry, Vectors.
We are asked to find the number of ways in which the time table can be completed if algebra and geometry are never immediate in the schedule.
The total number of ways in which the time table can be scheduled is:
Total number of subjects = 5
Number of ways = $5! = 120$
The total number of ways in which the time table can be scheduled such that algebra and geometry are always immediate.
Let’s consider algebra and geometry as one subject. Now the total number of subjects will be 4.
The total number of ways these 4 subjects can be arranged = $4! = 24$
Now the number of ways geometry and algebra can be arranged within themselves is $2! = 2$
Hence the total number of ways of arranging the 5 subjects such that geometry and algebra are always immediate is $24 \times 2 = 48$
Now to find the number of ways where algebra and geometry are never immediate is:
Total number of ways – number of ways such that algebra and geometry are always immediate
$120 - 48 = 72$

So, the correct answer is “Option A”.

Note: The two most important formulae to be remembered in permutations and combinations are:
When there are $n$ objects and all of them should be arranged, the total number of ways = $n!$
When there are $n$ objects and only $r$ of them are to be chosen, the total number of ways = $\dfrac{{n!}}{{(n - r)!}}$