UPSC NDA Math Model Paper 7

Given,a bag contains 3 black, 4 white and red balls.
The number of selections of at most 6 balls containing balls of all the colours = (The number of selection of a minimum of 3 balls with a different colour)(The number of selection of maximum ball with a different colour)
Now, the number of selection of a minimum of 3 balls with different colour = C(3,1) × C(4, 1) × C(2,1)
= 3 × 4 × 2 = 4!
The number of selection of maximum ball with different colour = C(6,3) + C(6, 2) + C(6,1) + C(6,1)
= 20 + 15 + 6 +1 = 42.
The number of selections of at most 6 balls containing balls of all the colours = 42 × 4!