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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.

**Complete step-by-step answer:**

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.

Now we can start arranging passengers using combinations step by step. But for that, you need to know the idea of combination. A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter.

$ \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 ${}^n{C_r} = {}^n{C_{n - r}}$

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