Question

The number of ways of arranging $n$ persons, if out of any two seats located symmetrically in the middle of the row at least one is empty is

• A. ${(}^{m/2}{C}_n\left)\right(2^n)-1$
• B. ${}^{m/2}{P}_n$
• C. ${(}^{m/2}{P}_n\left)\right(2^n-1)$
• D. ${(}^{m/2}{P}_n\left)\right(2^n)$

Answer 1

Answer: (D)

${(}^{m/2}{P}_n\left)\right(2^n)$

$m$ is even. Let $m=2k$, where $k$ is some positive integer. We can choose $n$ seats out of the $k$ seats to the left of the middle seat in ${}^k{C}_n$ ways. Each chosen seat can be either empty or occupied. Thus, the number of ways of choosing seats for $n$ persons is equal to ${(}^k{C}_n\left)\right(2^n)$. We can arrange $n$ persons at these seats in ${}^n{P}_n$ ways. Hence, the required number of arrangements is given by
$(n!){(}^k{C}_n\left)\right(2^n)={(}^k{P}_n)\left(2^n\right)={(}^{m/2}{P}_n\left)\right(2^n)$.