A box contains ‘x’ indigo, ‘x – 4’ violet and ‘x – 8’ blue balls. If two balls are drawn at random then probability of getting a violet ball and an indigo ball together is 1/3. If three balls are drawn from the box, then find the probability of getting three different coloured balls.

Total number of balls in the box = x – 4 + x – 8 + x = 3x – 12

According to question;
{ ^{(x – 4)} C 1 ×  ^x C 1 }/ ^{(3x – 12)} C 2  = 1/3
Or, 3 × {2 × (x – 4) × x} = (3x – 12) × (3x – 13)
Or, {2 × (x – 4) × x} = (x – 4) × (3x – 13)
Or, 2x ²  – 8x = 3x ²  – 25x + 52
Or, x ²  – 17x + 52 = 0
Or, x ²  – 13x – 4x + 52 = 0
Or, x(x – 13) – 4(x – 13) = 0
Or, (x – 4)(x – 13) = 0
Or, x = 13 or x = 4 (not possible)
Number of indigo balls = 13
Number of violet balls = 13 – 4 = 9
Number of blue balls = 13 – 8 = 5
Total number of balls = 3 × 13 – 12 = 27
Desired Probability = ( ¹³ C 1 ×  ⁹ C 1  × ⁵ C 1 )/ ²⁷ C 3  = (6 × 13 × 9 × 5)/(27 × 26 × 25) = 1/5