How many ways can 3 people can be seated in a row of 15 chairs?

Question

(1) Find the number of ways 3 people can be seated in a row of 15 chairs, explaining each step in turn.

(2) Use the binomial theorem to expand the following equation, explaining each step in turn:

(6x+3y)^4

Answer 1

(1) Out of the 15 chairs there are C(15, 3) ways of choosing 3 chairs where they can be seated. Out of the 3 chosen seats there are 3! ways to arrange seating order among the 3 students. Hence, the answer should be C(15, 3) × 3! = 15! / 12! = 15 × 14 × 13 = 2730.

(2) (6x + 3y)⁴ = C(4, 0) (6x)⁴ (3y)⁰ + C(4, 1) (6x)³ (3y)¹ + C(4, 2) (6x)² (3y)² + C(4, 3) (6x)¹ (3y)³ + C(4, 4) (6x)⁰ (3y)⁴ = 6⁴ • x⁴ + 4 • 6³ • 3 x³y + 6 • 6² • 3² x²y² + 4 • 6 • 3³ xy³ + 3⁴ • y⁴ = 1296x⁴ + 2592x³y + 1944x²y² + 648xy³ + 81y⁴