Photo to Logic to Math

That's a photo of a group of friends standing side by side in a row.

It reminds me a logic which leads to a small math-concept at the end.

Let us consider that some persons are standing in a row. Some clues are given and we need to find the number of persons.

Problem-1)Find the number of persons?

Clue: "In the row, a person called 'A' is standing second from left and fourth from right".

This clue is enough to answer the question. If we count from left,

A is second from left=> there are two persons up to A

A is fourth from right => there are three persons after A (as it is four including A)

So total number of persons = 2+3 = 5

It can be shown like this:

                                           *           A          *           *           *

Now let us go to a little complicated problem by introducing some more info.

Problem-2)Find the number of persons?

First Clue: “person 'A' is fourth from left”

Second Clue: “person 'B' is fifth from right”

Are these clues enough to get the answer? Obviously not...

Third Clue: There is one person in between A and B

Is it okay now? Can we solve the puzzle? Let us see...

"A is fourth from left" => there are four persons up to A (including A) from left-end.

"B is fifth from right" => there are five (including B) up to the right-end.

"There is one person in between A and B" => This guy shall be added to the count

So total number of persons = 4+5+1 = 10

It can be shown like this:

                         *           *           *           A          *           B          *           *           *           *

But is it okay?

No, there is some ambiguity in the third clue. Ambiguity is about the order that A and B stand in the row. If we count from left, we are not sure whether A comes before B or the other-way-round. In the scenario shown above (let us call it scenario-I), A comes before B. If B comes before A, then another scenario (call it scenario-II) may arise as shown below. Can't it satisfy all the three clues?

                                  *           B          *           A          *           *

Here again, "A is fourth from left" and "B is fifth from right" and "There is one person in between A and B".

And here there are only six persons in total.

My point is: For the second problem, the given three clues are not enough to answer the question straight away. This type of concept is useful in "Data Sufficiency" type of problems given in competitive exams.

Second problem takes me to "set theory" concept:

Scenario-I of second problem leads us to disjoint sets. One set with 4-persons (includes A) and other set with 5-persons (includes B) and there is one person not belonging to these two sets. We can consider third set for this guy. These three sets are disjoint to one another. 

n(X) = 4

n(Y) = 5

n(Z) = 1

Total number of persons = n(XUYUZ) = n(X)+n(Y)+n(Z) = 4+5+1 = 10

"n(XUYUZ) = n(X)+n(Y)+n(Z)" -this formula holds good for disjoint sets.

Scenario-II leads us to two intersecting sets. One set with 4-persons (includes B as well) and

other set with 5-persons (includes A as well) and there is an intersection of these two sets with 3-persons in common (A, B and the other guy).

n(X) = 4

n(Y) = 5

n(X∩Y) = 3

Total number of persons = n(XUY) = n(X)+n(Y)-n(X∩Y) = 4+5-3 = 6
"n(XUY) = n(X)+n(Y)-n(X∩Y)" -this formula holds good for intersecting sets.