Question

There are N men sitting around a circular table at N distinct points. Every possible pair of men except the ones sitting adjacent to each other sings a 2 minute song one pair after other. If the total time taken is 88 minutes, then what is the value of N?
A)8 B) 9
C) 10
D) 11​

Answer 1

Given :

There are N men sitting around a circular table at N distinct points

Every possible pair of men except the ones sitting adjacent to each other sings a 2 minute song one pair after other .

To time taken to sing songs  from one pair to another = 88 minutes

To Find :

The value of N

Solution :

When we join these N men around a circular table , then a polygon is formed

A person cannot make pair with himself and his adjacent men

∴ we can not count 3  men out of N

So, one men make pair with = ( N - 3 ) men

∴ Total men counted = N ( N - 3 )

But , Each combination is counted twice

i.e From one pair to another and vice-versa

So, Total men counted = $\dfrac{N ( N - 3 )}{2}$

Again

∵  Total time for song singing from one pair to another = 88 min

And Each pair sings for 2 min

∴, Time for each pair =  $\dfrac{88 min}{2 min}$

                                   = 44 min

Now,

Total men counted =  Time for each pair

i.e  $\dfrac{N ( N - 3 )}{2}$ = 44

Or,  N ( N - 3 ) = 44 × 2

Or,  N ( N - 3 ) = 88

We can write

      N ( N - 3 ) = 11 × 8

Or,  N ( N - 3 ) = 11 × ( 11 - 3 )

Thus after comparing , its is clear that , N = 11

Hence, The value of N for the given condition is 11  . Answer