Question

In how many ways can 4 people be seated in a row containing 6 seats?

Answer 1

Hint:As given in the question we have to choose 4 seats from the given 6 seats and then we can place 4 people in those seats and then we will have to arrange them. We will use the formula ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$ of permutation to solve this question.

Complete step-by-step answer:
Let’s start our solution,
Permutation means choosing the required number of terms and then arranging them.
Now if we have n different objects and from them we need to pick r objects and then the formula is ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
First we will choose 4 seats from the 6 given seats and then arrange the 4 people.
Here we have n = 6 which is the total number of seats and r = 4.
Now we will use the formula of combinations${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$, where n = 6, r = 4.
$\begin{align}
  & =\dfrac{6!}{\left( 6-4 \right)!} \\
 & =\dfrac{6\times 5\times 4\times 3\times 2}{2!} \\
 & =6\times 5\times 4\times 3 \\
 & =360 \\
\end{align}$
Hence the number of ways 4 people be seated in a row containing 6 seats is 360.

Note:The formula that we have used to solve this question is ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$. One can also look at it in this way that the first person has 6 options, the second person has 5 options and so on, and from this also we will get the same answer. So we can obtain the total number of seats by multiplying the possibilities.