My friend Asha, Neesha, Vijay, Parveen and Seema and Myself,
we decided to have a group photo taken in the studio. We decided to sit in a
row. How many different arrangements can be made of the order in which we could
have sat?
After the sitting at the photo studio, we all decided to
lunch together in a restaurant.
The waiter led us to a round table. We had a little bit of
an argument about who should sit next to whom.
How many different arrangements can be made of the order in
which we could have sat?
Answer: (a) 720. – It may surprise you to see such a
big number of arrangements. But it is the product of 6x5x4x3x2x1 that is 6! Or
6- factorial 6.
Here, for example the lefthand lady can be any one of them,
so there are 6 ways of choosing her.
The next lady from the lefthand side can be chosen in 5 ways
from the remaining 5 ladies.
The next lady in 4 ways from the remaining 4 and the next
lady 3 ways and so on.
If there was only one more lady making us 7 ladies together,
the number of possible arrangements would be 7 or 5040.
If there were 9 ladies then there would be more than three
hundred thousand ways of arranging us.
(b) 120. Here the situation is entirely different. In this case
the answer is not the same as in the case of (a), because it is only the order
which is considered here and not the actual position.
In this case there will be 6 positions in which the same
order will be found but each position will be turned round relatively to the
other.
And there is another way of considering this problem. This
is to keep one lady always in the same place and then arrange the remaining 5
ladies. This car is done in 5 ways or 120.
Any order arranged clock-wise has an equivalent order arranged
anti-clockwise. So the number of 120 different ways includes both these as
separate arrangements.
