A fair six-sided die is rolled 4 times. The probability that exactly two distinct numbers appear across the four rolls is:

Step-by-step solution

The total number of possible outcomes when a die is rolled 4 times is 6^4 = 1296 . For exactly two distinct numbers to appear, we first select the two numbers from the six available numbers on the die. This can be done in ⁶C₂ = 15 ways. Next, we need to form a sequence of 4 rolls using exactly these two selected numbers such that both numbers appear at least once. The total number of ways to assign the 4 rolls to the two numbers is 2^4 = 16 . However, this includes the 2 cases where all four rolls show the same num