A box contains 6 bottles of variety 1 drink, 3 bottles of variety 2 drink, and 4 bottles of variety 3 drink. Three bottles of them are drawn at random, what is the probability that the three are not of the same variety?

Total number of drink bottles = 6 + 3 + 4 = 13.
Let S be the sample space
Then, n(S) = number of ways of taking 3 drink bottles out of 13.
Therefore, n(S) = ¹³C₃
= (13 x 12 x 11)/(1 x 2 x 3)
= 66 x 13
= 858.
Let E be the event of taking 3 bottles of the same variety
Then, E = event of taking (3 bottles out of 6) or (3 bottles out of 3) or (3 bottles out of 4)
n(E) = ⁶C₃ + ³C₃ + ⁴C₃
= (6 x 5 x 4 )/ (1 x 2 x 3) + 1 + (4 x 3 x 2) / (1 x 2 x 3)
= 20 + 1 + 4
= 25
The probability of taking 3 bottles of the same variety = n(E)/n(S)
= 25/858.
Then, the probability of taking 3 bottles are not of the same variety = 1 - 25/858
= 833/858.