Question

A table has a provision for $7$ seats, $4$ being on one side facing the window and $3$ being on the opposite side. The number of ways in which $7$ people can be seated at the table if $2$ people, $X$ and $Y$ must sit on the same side is

• A. $3260$
• B. $2160$
• C. $3350$
• D. None of these

Answer 1

Answer: (B)

$2160$

1) When $X$ and $Y$ sit on the side facing the window- first we need to select $2$ persons for the side facing the window and $3$ for the opposite side from the remaining $5$.

Thus number of selections are ${{}^{5}C}_1\times {{}^{3}C}_3$

For each of the selection, the arrangements are $=4!\times 3!$

Thus the number of ways to sit when $X$ and $Y$ are on the side facing the window are ${{}^{5}C}_1\times {{}^{3}C}_3\times 4!\times 3!$

2) When $X$ and $Y$ are on the other side- as before the number of ways to sit are ${{}^{5}C}_1\times {{}^{4}C}_4\times 4!\times 3!$

Thus the required number of ways ${{}^{5}C}_1\times {{}^{3}C}_3\times 4!\times 3!+{{}^{5}C}_1\times {{}^{4}C}_4\times 4!\times 3!=2160$