Question

Five friends $ {F_1},{F_2},{F_3},{F_{4,}}{F_5} $ book five seats $ {C_1},{C_2},{C_3},{C_{4,}}{C_5} $ respectively of movie KABIL independently (i.e. $ {F_1} $ books $ {C_1} $ , $ {F_2} $ books $ {C_2} $ and so on). In how many different ways they can sit on these seats if $ {F_1} $ and $ {F_2} $ wants to sit adjacent to each other.

Answer 1

Hint: When $ n $ objects are to be placed in $ r $ seats randomly, then this can be done in $ {n^r} $ ways. Use the formula $ {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $ in this question to reduce the complication of this question. Also, the number of ways of arranging $ n $ things is in $ n! $ ways, this formula will help to find out the number of ways to arrange things.

Complete step-by-step answer:
Five friends $ {F_1},{F_2},{F_3},{F_{4,}}{F_5} $ book five seats $ {C_1},{C_2},{C_3},{C_{4,}}{C_5} $ respectively for a movie.
These 5 friends can sit on any of the 5 seats they booked.
It is known that there are $ n! $ ways to arrange $ n $ objects. Similarly, 5 friends can be seated in $ 5! $ ways.
  $ r $ objects can be chosen from $ n $ objects in $ {}^n{C_r} $ ways, where $ {}^n{C_r} = \dfrac{{n!}}{{r! \cdot \left( {n - r} \right)!}} $ .
But, two friends $ {F_1},{F_2} $ wants to sit together.
So first we have to choose 2 seats for $ {F_1},{F_2} $ and this can be done in $ {}^5{C_2} $ ways.
The other 3 friends can now be seated in $ 3! $ . All the friends are too seated for the movie together so we have to multiply all the factors.
The different ways 5 friends can sit on these seats if $ {F_1} $ and $ {F_2} $ wants to sit adjacent to each other is given by $ {}^5{C_2} \cdot 3! $ .
  $
\Rightarrow {}^5{C_2} \cdot 3! = \dfrac{{5!}}{{2!\left( {5 - 2} \right)!}} \cdot 3!\\
 = \dfrac{{5!}}{{2! \cdot 3!}} \cdot 3!\\
 = \dfrac{{5!}}{{2!}}
  $
On simplifying, we get,
  $
\Rightarrow {}^5{C_2} \cdot 3! = \dfrac{{120}}{2}\\
 = 60
  $
5 friends can sit on 5 booked seats if $ {F_1} $ and $ {F_2} $ wants to sit adjacent to each other in 60 ways.

Note: Students must avoid mistakes while using the formula for calculating the number of ways when $ r $ objects can be chosen from $ n $ objects in $ {}^n{C_r} $ ways. Instead of taking the number of ways as $ {}^n{C_r} $ , students can take it mistakenly as $ {}^r{C_n} $ . Also, the language of the question should be understood in a clear manner so that conditions are well expressed in terms of mathematical symbols.