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# A double-decker can accommodate $20$ passenger, $7$ in the lower deck and $13$ in the upper deck. Find the number of ways the passengers can be accommodated if $5$ want to sit only in the lower deck and $8$ want to sit only in the upper deck.

Last updated date: 02nd Aug 2024
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Hint:Remember you don’t have to arrange the passengers who are already sitting or have a preference. Arrange the remaining passengers using combinations in the remaining seats in the lower or upper deck.

Let’s first try to analyse the question properly. In the given problem, there are total $20$ passengers that are to be arranged in $20$ seats of a double-decker that has $7$ seats in the lower deck and $13$ in the upper deck. But we also have a condition that, $5$ passenger should be arranged only in the lower deck and $8$ passengers should be arranged only in the upper deck.
So basically, $5$ passengers in $7$ seats of the lower deck are already arranged then $8$ passengers in $13$ seats of the upper deck are already arranged. And then remaining $7$ passengers in $7$ remaining seats.
$\Rightarrow$ If $'n'$ is the number of things to choose from, and we choose $'r'$ of them, no repetition, the order doesn't matter; then we can represent then as: ${}^n{C_r} = \left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right) = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Out of $7$ remaining passengers, we have to arrange them in remaining $2$ seats in the lower deck and in $5$ seats in the upper deck.
$\Rightarrow$ We can arrange remaining passengers in the upper deck then after those remaining two passengers will be arranged in remaining two seats by itself $\Rightarrow {}^7{C_5} = \dfrac{{7!}}{{5!\left( {7 - 5} \right)!}}$
Therefore, the number of required ways$= \dfrac{{7!}}{{5!\left( {7 - 5} \right)!}} = \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7}}{{1 \times 2 \times 3 \times 4 \times 5 \times 1 \times 2}} = \dfrac{{6 \times 7}}{{1 \times 2}} = 21$
Hence, the number of ways to arrange the passengers as required is $21$.
Note:Try to visualize the given condition properly before starting a solution. An alternative approach can be taken by arranging the remaining passengers in the lower deck first, i.e. by using${}^7{C_2}$. But you will find the same answer since ${}^n{C_r} = {}^n{C_{n - r}}$