Question

If 12 persons are seated in a row, the number of ways of selecting 3 persons from them, so that no two of them are seated next to each other is

• A. $85$
• B. $100$
• C. $120$
• D. $240$

Answer 1

The correct option is C $120$

Given that 12 persons are sitting in a row,

Let us choose 3 persons in a random way,so the remaining persons are 9,

let $w$ be the number of persons sitting left to the first person,

let $x$ be the number of persons sitting in between the first and second person,

Let $y$ be the number of persons sitting between the second and third person,

Let $z$ be the number of persons sitting right to the third person,

so,$w+x+y+z=9$,

but no two persons should be next to each other,

$⟹y,z\ge 1$,

let $y=a+1$ and $z=b+1$,

The equation now turns to,

$w+a+b+z=7$,

$\therefore$ the number of ways of selecting 3 persons such that no 2 are sitting next to each other is the number of solutions to the above equation,

i.e.,$\left(\begin{array}{c}7+4-1\\ 4-1\end{array}\right)=\left(\begin{array}{c}10\\ 3\end{array}\right)=120$