Question

Eighteen quests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three on the other side. Determine the number of ways in which the seating arrangement can be made? 

(a) ¹⁸C₄ × ¹⁴C₃ × 9 ! × 2 

(b) ¹¹C₅ × ⁶C₆ × 9 ! × 2 ! 

(c) ¹¹C₅ × ⁶C₆ × 9 ! × 9 ! 

(d) ¹⁸C₄ × ¹⁴C₃ × 9 ! × 2 !

Answer 1

(c) ¹¹C₅ × ⁶C₆ × 9! × 9!

As four particular guests want to sit on a particular side say (S₁) and three others on the other side, say, (S₂). So we are left with 11 guests out of which we choose 5 for side S₁ from remaining 11 in ¹¹C₅ ways and 6 from remaining 6 for side S₂ in ⁶C₆ ways. 

Also the 9 persons on each side can be arranged amongst themselves in 9! ways. 

∴ Required number of ways of making the seating arrangement = ¹¹C₅ × ⁶C₆ × 9! × 9!