Three different people are to be selected at random, and each will be given one gift card. There is one card from Home Depot, one from Best Buy, and one from Red Lobster. The first person selected gets to choose one of the cards. The second person selected gets to choose between the two remaining cards. The third person selected gets the third card.

A) Determine the number of points in the sample space.
B) Construct a tree diagram and determine the sample space.

Determine the probability that
C) The Best Buy card is selected first
D) The Home Depot card is selected first and the Red Lobster card is selected last.
E) The cards are selected in this order: Best Buy, Red Lobster, Home Depot.

A) To determine the number of points in the sample space, we need to consider all the possible outcomes. In this case, there are 3 different people who can be selected first, then 2 people remaining who can be selected second, and finally, the remaining 1 person is selected last. Therefore, the sample space contains 3 x 2 x 1 = 6 possible outcomes.

B) To construct a tree diagram, we start with the first person being selected. We have 3 options: Home Depot, Best Buy, or Red Lobster. From each of these options, we draw branches for the second person selected, with 2 options remaining. Finally, for the third person selected, we have only 1 option remaining. The tree diagram should look like this:

Home Depot
/ \
Best Buy Red Lobster
| |
Best Buy Red Lobster
| |
Red Lobster Best Buy
| |
Red Lobster Home Depot
| |
Best Buy Home Depot
| |
Home Depot Best Buy

C) The probability that the Best Buy card is selected first can be calculated by dividing the number of favorable outcomes (where Best Buy is selected first) by the total number of possible outcomes. From the tree diagram, we can see that there are 2 favorable outcomes ("Best Buy Red Lobster" and "Best Buy Home Depot") out of 6 total outcomes. Therefore, the probability is 2/6 = 1/3.

D) The probability that the Home Depot card is selected first and the Red Lobster card is selected last can be calculated in a similar manner. From the tree diagram, we can see that there is only 1 favorable outcome ("Home Depot Best Buy") out of 6 total outcomes. Therefore, the probability is 1/6.

E) The probability that the cards are selected in the order Best Buy, Red Lobster, Home Depot can also be calculated using the tree diagram. We can see that there is only 1 favorable outcome ("Best Buy Red Lobster Home Depot") out of 6 total outcomes. Therefore, the probability is 1/6.