Question

Passengers are to travel by a double decked bus which can accommodate $13$ in the upper deck and $7$ in the lower deck. The number of ways that they can be distributed if $5$ refuse to sit in the upper deck and $8$ refuse to sit in the lower deck is-

• A. $25$
• B. $21$
• C. $18$
• D. $15$

Answer 1

The correct option is B

$21$

Explanation for the correct option:

Finding the required number of ways of sitting arrangements:

Given that,

$13$ passengers can be accommodated on the upper deck and $7$ in the lower deck

$5$ passengers refused to sit on the upper deck

Also Given that,

we have $7$ seats in the lower deck that can be taken by $5$ passengers in ${}^7{C}_5$ ways

Now, there are $2$passengers and $2$ seats left, so they can be seated in ${}^2{C}_2$ways

Therefore, the total number of ways is

$$\begin{gathered} {}^7{C}_{5}x{}^2{C}_2 \\ =\frac{7!}{5!\times 2!}\times \frac{2!}{2!\times 0!}\left[{\because }^n{C}_r=\frac{n!}{r!(n-r)!}\right] \\ =21\times 1 \\ =21 \end{gathered}$$

Therefore, the correct answer is option (B).