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Question: There are 5 red, 4 white, and 3 blue balls in a box. If 3 balls are drawn at random from the box, what is the probability that all three are red?

Solution:
Total number of balls = 5 + 4 + 3 = 12 balls
We are drawing 3 balls at random without replacement.
Total number of ways to choose 3 balls from 12
= ¹²C₃
= 12!/(3! × 9!)
= (12 × 11 × 10)/(3 × 2 × 1)
= 220

And
Number of favorable ways (all 3 balls are red)
= Number of ways to choose 3 red balls from 5 red balls
= ⁵C₃
= 5!/(3! × 2!)
= (5 × 4 × 3)/(3 × 2 × 1)
= 10

∴ Probability = (favorable outcomes)/(total outcomes)
= 10/220
= 1/22
So the probability that all three balls are red is 1/22.

Alternatively, using sequential probability (without replacement):
P(1st Red) = 5/12
P(2nd Red) = 4/11 (4 red left out of 11 total)
P(3rd Red) = 3/10 (3 red left out of 10 total)

∴ Total Probability = (5/12) × (4/11) × (3/10)
= 60/1320
= 1/22.