No of selections and probability

Six balls are selected from a set of 44 balls, numbered from 1 to 44. The balls are colored. The balles numbered, 1, 5, 9 ... are red, 2,6,10 are orange, the balls no, 3 ,7, 11 are yellow and the balls numbered 4,8,12... are pink

What is the probability that the selected six balls contain:

f)Four colours:

Heres my attempt: there are two cases:

[1,1,1,3] and [1,1,2,2]

This equates to

(4C3 x (11C1)^3 x (11C3) + 4C2 x (11C1)^2 x (11C2)^2)/44C11

However the book has a different solution:

(4C3 x (11C1)^3 x (11C3) + 4C2 x** 2C1 **X (11C1)^2 x (11C2)^2)/44C11

Can someone explain to me?