Given below are two quantities named A and B. Based on the given information, you have to determine the relation between the two quantities. You should use the given data and your knowledge of Mathematics to choose between the possible answers. 6 senior managers and 4 junior managers need to be seated in a round table for a meeting. Quantity A: Number of ways in which no two junior managers sit together Quantity B: Number of ways in which all senior managers sit diagonally opposite to each other?

Option 5 :  Quantity A = Quantity B (or) No relation can be obtained

Quantity A:

Let the seat be numbered 1, 2, 3, …, 10

Only way two junior managers not sitting together is that all the managers get to sit in between two senior managers

Five senior managers are chosen and made to sit in alternative chairs. Then 4 junior managers and 1 more senior manager sits in the remaining chairs

⇒ Number of ways of choosing five senior managers from six senior managers = ⁶C₅

⇒ 6!/(5! × 1!) = 6

There are two ways of making the selected senior managers to sit in the alternative seats: 1, 3, 5, 7 and 9 or 2, 4, 6, 8 and 10

⇒ Number of ways in which the selected senior managers arrange themselves in the 5 seats = ⁵P₅ = 5!

⇒ Number of ways in which the remaining 5 managers arrange themselves in 5 seats = 5!

⇒ Total number of ways in which two junior managers never sit together = 6 × 5! × 2 × 5! = 172800

Quantity B:

⇒ Number of diagonally opposite seat sets in the table = 5

⇒ Number of senior managers = 6

Number of ways to make 3 sets of senior managers to sit diagonally opposite to each other = Number of ways of selecting 2 managers out of 6 × Number of ways selecting 2 managers out of 4 × Number of ways of selecting two managers out of 2

⇒ (⁶C₂ ⁴C₂ ²C₂)/3! = 15

⇒ Number of ways to select three diagonally opposite seats from 5 sets = ⁵C₃ = 10

⇒ Number of ways of arranging 3 sets of people in 3 sets of seats = 3!

In each set, two of them can shift among themselves

⇒ 2 × 2 × 2 = 8 seating arrangements with in themselves.

⇒ Number of ways in which 4 junior managers is seated in the remaining four seats = 4!

⇒ Total number of ways = 15 × 10 × 3! × 8 × 4! = 172800

∴ Quantity A = Quantity B