Question

1. A person has 8 friends, of whom 5 will be invited to a party. How many choices are there if and only if 2 of the friends will only attend together? 2. Suppose that 5 of the integers 1,2,…,15 are to be chosen randomly with equal chances. Find the probability that 8 is the third smallest value chosen.

Answer 1

In the first question, there are 20 choices for the friends to attend the party. In the second question, the probability of 8 being the third smallest value chosen is determined using combinations.

Step-by-step explanation:

In the first question, there are 8 friends and 5 of them will be invited to a party. However, 2 friends must attend together. To calculate the number of choices, we need to consider that the 2 friends who attend together can be chosen in 1 way. The remaining 3 friends can be chosen from the remaining 6 friends in 6 choose 3 ways, which is equal to 20. Therefore, there are 1 * 20 = 20 choices.

In the second question, we are choosing 5 integers randomly from the numbers 1 to 15. There are a total of 15 choose 5 ways to choose 5 integers from 1 to 15. To determine the probability that 8 is the third smallest value chosen, we need to choose 2 integers smaller than 8 and 2 integers larger than 8. There are 7 choose 2 ways to choose 2 integers smaller than 8 and 6 choose 2 ways to choose 2 integers larger than 8. Therefore, the probability is (7 choose 2) * (6 choose 2) divided by (15 choose 5).