Question

Find the number of ways in which 16 constables can be assigned to patrol 8 villages 2 each?

Answer 1

Hint: The number of ways in which 16 constables can be assigned to patrol 8 villages, 2 each is calculated by selecting 2 constables out of 16 constables using combinatorial approach then we multiply this selection with selecting 2 constables out of 14 constables likewise we multiply till we can select the last pair of constables in the last village.

Complete step by step answer:
We have given 16 constables and are asked to find the ways in which we can assign these 16 constables in 8 villages with 2 constables in each village. These numbers of ways can be calculated by selecting 2 constables out of 16 constables using the combinatorial approach for village first.
${}^{16}{{C}_{2}}$
Now, we are going to select 2 constables out of 14 constables for village second and multiplying with the above selection and we get,
${}^{16}{{C}_{2}}\times {}^{14}{{C}_{2}}$
Selecting 2 constables out of 12 constables for village third and multiplying this selection with the above and we get,
${}^{16}{{C}_{2}}\times {}^{14}{{C}_{2}}\times {}^{12}{{C}_{2}}$
Selecting 2 constables out of 10 constables for village fourth and multiplying this selection with the above and we get,
${}^{16}{{C}_{2}}\times {}^{14}{{C}_{2}}\times {}^{12}{{C}_{2}}\times {}^{10}{{C}_{2}}$
This multiplication will keep going on till we reach the last village and the expression will look like as follows:
${}^{16}{{C}_{2}}\times {}^{14}{{C}_{2}}\times {}^{12}{{C}_{2}}\times {}^{10}{{C}_{2}}\times {}^{8}{{C}_{2}}\times {}^{6}{{C}_{2}}\times {}^{4}{{C}_{2}}\times {}^{2}{{C}_{2}}$
Writing the above expression in factorial form and we get,
$\begin{align}
  & \dfrac{16!}{2!14!}\times \dfrac{14!}{2!12!}\times \dfrac{12!}{2!10!}\times \dfrac{10!}{2!8!}\times \dfrac{8!}{2!6!}\times \dfrac{6!}{2!4!}\times \dfrac{4!}{2!2!}\times \dfrac{2!}{2!0!} \\
 & =\dfrac{16!}{{{\left( 2! \right)}^{8}}0!} \\
\end{align}$
We know that the value of $0!=1$ so substituting this value in the above expression and we get,
$\begin{align}
  & \dfrac{16!}{{{\left( 2.1 \right)}^{8}}} \\
 & =\dfrac{16!}{{{2}^{8}}} \\
\end{align}$
From the above solution, we have found the number of ways in which 16 constables can be assigned to patrol 8 villages with 2 each.

Note: To solve the above problem, you must know how to write a combinatorial approach. Also, you should know what the expansion of ${}^{n}{{C}_{r}}$is. Along with that, you must know what the value of $0!$ is. You might be wondering why we haven’t solved the final answer completely, the reason is the solution will be very long then and even in examination also you will find the options of the above form.