Final answer:
If the first, third, and fourth numbers must be the same, there are 36 different sequences possible. A fair six-sided die has 6^n different sequences possible when rolled n times. The number of sequences consisting entirely of even numbers is (1/2)^n * 6^n.
Step-by-step explanation:
A fair six-sided die has six possible outcomes: 1, 2, 3, 4, 5, or 6. If the die is rolled n times, the number of different sequences of outcomes is 6^n. For example, if the die is rolled 2 times, there are 6^2 = 36 different sequences possible.
If we want to find the number of sequences consisting entirely of even numbers, we need to consider that there are 3 even numbers on the die: 2, 4, and 6. So, for each roll, there is a 1/2 chance of getting an even number. If the die is rolled n times, the probability of getting an even number on each roll is (1/2)^n. Therefore, the number of sequences consisting entirely of even numbers is (1/2)^n * 6^n.
If the first, third, and fourth numbers must be the same, we have fixed outcomes for those positions. The second and fifth positions can have any of the 6 possible outcomes. So, the number of sequences in this case is 6^2 = 36.