a. To find the probability that Comic B will perform second, we need to consider the total number of possible orders of performance. Since there are 6 comics, there are 6 choices for the first performer. After the first performer is chosen, there are 5 remaining choices for the second performer. Therefore, the total number of possible orders of performance is 6 * 5 = 30.
Since we want Comic B to perform second, there is only 1 specific order that satisfies this condition. Therefore, the probability that Comic B will perform second is 1 out of 30, or 1/30.
b. To find the probability that Comic D will perform fifth and Comic A will perform first, we need to consider the total number of possible orders of performance. As mentioned before, there are 6 comics, so there are 6 choices for the first performer. After the first performer is chosen, there are 5 remaining choices for the second performer, and so on. Therefore, the total number of possible orders of performance is 6 * 5 * 4 * 3 * 2 * 1 = 720.
Since we want Comic D to perform fifth and Comic A to perform first, there is only 1 specific order that satisfies this condition. Therefore, the probability that Comic D will perform fifth and Comic A will perform first is 1 out of 720, or 1/720.
c. To find the probability that the comedians will perform in the order B, D, A, E, C, F, we need to consider the total number of possible orders of performance, which is the same as before, 720.
Since we want the specific order B, D, A, E, C, F, there is only 1 specific order that satisfies this condition. Therefore, the probability that the comedians will perform in the given order is 1 out of 720, or 1/720.
d. To find the probability that Comic C or Comic F will perform third, we need to consider the total number of possible orders of performance, which is still 720.
To determine the number of orders in which Comic C or Comic F will perform third, we need to consider two cases:
1. Comic C performs third: In this case, after choosing the first performer and the second performer, there are 4 remaining comics, including Comic C and Comic F. Therefore, there are 4 choices for the third performer, and then 3 choices for the fourth performer, and so on. This results in a total of 6 * 5 * 4 * 3 * 2 * 1 = 720 possible orders.
2. Comic F performs third: Similar to the previous case, after choosing the first performer and the second performer, there are 4 remaining comics, including Comic C and Comic F. Therefore, there are 4 choices for the third performer, and then 3 choices for the fourth performer, and so on. This also results in a total of 6 * 5 * 4 * 3 * 2 * 1 = 720 possible orders.
Since we are considering two separate cases, we need to add up the number of orders in each case. Therefore, the total number of orders where Comic C or Comic F will perform third is 720 + 720 = 1440.
Therefore, the probability that Comic C or Comic F will perform third is 1440 out of 720, or 1440/720, which simplifies to 2.