Four fair dice D₁, D₂, D₃ and D₄ ; each having six faces numbered 1,2,3,4,5 and 6 are rolled simultaneously. The probability that D₄ shows a number appearing on one of D₁, D₂ and D ₃ is

Step-by-step solution

D₄ can show a number appearing on one of D₁, D₂ and D₃ in the following cases. Case I : D₄ shows a number which is shown by exactly one of D₁, D₂ and D₃ . D₄ shows a number in 6 1 ways. One out of D₁, D₂ and D₃ can be selected in 3 1 ways. [The selected die shows the same number as on D₄ in one way and rest two dice show the different number in 5 ways each, ] Number of ways = 6 1 3 1 1 5 5=450 Case II : D₄ shows a number which is shown by exactly two of D₁, D₂ and D₃ . Number of ways = 6 1 3 2 1 1 5=90 Case III : D