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

The number of ways she can paint her 6 rooms is = 57

Note :The number of ways she can paint her 6 rooms is explained with the table below

	grey	Blue	Number of ways
	6	0	6C0=1
	5	1	6C1=6
	4	2	6C2=15
	3	3	6C3=20
	2	4	6C4=15
total			57

If she paints all room grey , it can be done in one way only( 6C0 =1 )

If she paints one room blue and remaining other grey , the number of ways she can pick that blue room is 6C1 =6

If she paints two rooms blue and remaining other grey , the number of ways she can pick that 2 blue room is 6C2 =15

If she paints three rooms blue and remaining other grey , the number of ways she can pick that 3 blue room is 6C3 =20

If she paints four rooms blue and remaining other grey , the number of ways she can pick that 4 blue room is  6C4 =15

Note : She cannot paint more than 4 blue rooms

Thus total number of ways =1+ 6+15+20+15 = 57