To calculate the number of possible outcomes for these 5 rolls, we need to consider the number of choices for each roll. Since you are rolling a 16-sided die 5 times, each roll has 16 possible outcomes. Therefore, the total number of outcomes for the 5 rolls can be calculated by multiplying the number of outcomes for each roll:
a) Number of possible outcomes = (Number of outcomes per roll) ^ (Number of rolls)
Number of possible outcomes = 16^5 = 1048576
So, there are 1,048,576 possible outcomes for these 5 rolls.
For part b, we need to determine how many of these outcomes produce five different numbers. Since each roll has 16 possible outcomes, the first roll can be any number from 1 to 16. After that, the second roll cannot be the same as the first roll, so it has 15 possible outcomes. Similarly, the third roll has 14 possible outcomes, the fourth roll has 13 possible outcomes, and the fifth roll has 12 possible outcomes. Therefore, the number of outcomes with five different numbers can be calculated as:
b) Number of outcomes with five different numbers = 16 * 15 * 14 * 13 * 12 = 524,160
So, there are 524,160 outcomes in which the 5 rolls produce five different numbers.
Finally, for part c, we need to calculate the probability that at least 2 of the rolls are the same. The opposite of "at least 2 of the rolls are the same" is "all rolls are different." We have already calculated the number of outcomes with five different numbers in part b, which is 524,160. Therefore, the probability that at least 2 of the rolls are the same can be calculated as:
c) Probability that at least 2 rolls are the same = 1 - (Number of outcomes with all rolls different / Total number of outcomes)
Probability that at least 2 rolls are the same = 1 - (524,160 / 1,048,576)
Probability that at least 2 rolls are the same = 0.4999 (rounded to 4 decimal places)
Therefore, the probability that at least 2 of the rolls are the same is approximately 0.4999 or 49.99%.