Question

There are 10 buses running between Delhi and Agra. In how many ways can a man go from Delhi to Agra and return by a different bus?

Answer 1

Hint:As we have 10 different buses so first we have to choose 1 from these 10 buses and then for the return journey we have to choose 1 one from the remaining 9 buses as we can’t choose the same bus for the return. Hence, we will use the formula for combination to choose the bus from the 10 buses.

Complete step-by-step answer:
Let’s start our solution,
Combination means that in how many ways we can choose from the given number of objects.Now if we have n different objects and from them we need to pick r objects, then the formula is
${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$
Now in the question we have given n = 10 which is the number of buses available and from that we have to choose 1 bus, r = 1.
Hence, using the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$ we get,
$\begin{align}
  & =\dfrac{10!}{\left( 10-1 \right)!1!} \\
 & =\dfrac{10\times 9!}{9!} \\
 & =10 \\
\end{align}$
Now for the return we have to choose 1 bus from 9 remaining buses,
Therefore, using the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$ for n = 9 and r = 1 we get,
$\begin{align}
  & =\dfrac{9!}{1!\left( 9-1 \right)!} \\
 & =\dfrac{9\times 8!}{8!} \\
 & =9 \\
\end{align}$
Now we have to multiply the two numbers to get the total number of possible ways that a man can go from Delhi to Agra and return by a different bus.
Hence, we get $9\times 10=90$

Note:The formula ${}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!r!}$ must be kept in mind. One thing that a student must understand is that we have to multiply the two possible cases because for every 1 bus he takes he has 9 options to return and so like that we get 10 times 9 which is 90.