Question

There is n number of seats and m number of people having to be seated in those n seats. How many possible arrangements are possible, if we are given m is less than n.

Answer 1

Hint: Now we know that the number of ways in which r objects can be placed in n places is given by \[^{n}{{C}_{r}}\] . Now after placing m people we can arrange them in m! ways. Hence the total number of arrangement is \[^{n}{{C}_{r}}m!\] .

Complete step-by-step solution:
Now in a given room, we gave n number of seats.
Now we want to place n people in those n seats. Hence to do so we will first select m seats from n.
Now we know that the number of ways of selecting r objects from n objects is $^{n}{{C}_{r}}$ .
Where $^{n}{{C}_{r}}=\dfrac{n!}{(n-r)!r!}$ and $a!=a\times (a-1)\times (a-2)\times ....\times (2)\times 1$
Hence we know that the number of ways of selecting m seats from n seats is given by.
$^{n}{{C}_{m}}$ . …………………… (1)
To understand this with an example
Consider 3 chairs numbered 1, 2, 3.
Now we want to select 2 chairs. Hence we can choose (1, 2), (2, 3), (1, 3).
Hence we have 3 selections
Similarly if we choose it by formula we have $^{3}{{C}_{2}}=\dfrac{3!}{2!(3-2)!}=\dfrac{3\times 2}{2\times 1(1)!}=3$
Now once we have selected m seats we will place m people on these seats.
This is the same as arranging m objects. Now we know that the total arrangements of m objects is given by m!
Hence the number of ways in which we can place m people on m seats is m! ………………………(2)
Hence from equation (1) and equation (2) we get the number of total arrangement is
$^{n}{{C}_{m}}m!$
Now we know that \[^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}\]
Hence we have
$\begin{align}
  & ^{n}{{C}_{m}}m!=\dfrac{n!}{\left( n-m \right)!m!}\times m! \\
 & ^{n}{{C}_{m}}m!=\dfrac{n!}{\left( n-m \right)!} \\
\end{align}$
Hence we have the total number of possible arrangements is $\dfrac{n!}{\left( n-m \right)!}$

Note: Now we have that number of arrangements of r objects in n places is given by $^{n}{{P}_{r}}$ where $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$ . hence using this we can directly find the number of ways to place m people in n chairs.