Question

Liza has a supply of blue paint and grey paint. She plans to paint each of the 6 rooms in her house with one of the two colours. She has enough grey paint to paint the entire house, but only enough blue paint for 4 rooms. In how many different ways can she paint the rooms of her house? For example, she may paint all the rooms grey, or she may paint half grey and half blue, etc. But she can't paint any more than 4 rooms blue.

Answer 1

Ways for X = 0 implies 0 blue rooms = C(6, 6) = 1

Ways for X = 1 implies 1 blue room = C(6,1)*C(5,5) = 6*1 = 6

Ways for X = 2 implies 2 blue room = C(6,2)*C(4,4) = 15*1 = 15

Ways for X = 3 implies 3 blue rooms = C(6, 3) * C(3,3) = 20

Ways for X = 4 implies 4 blue rooms = C(6, 4)* C(2, 2) = 15

Total ways = 1 + 6 + 15 + 20 + 15 = 57

Hence, total ways to paint the rooms of her house = 57

Please comment if any doubt. Thank you.