If F is not selected to attend the retirement dinner, then exactly how many different groups of four are there each of which would be an acceptable selection?
If F is not selected to attend the retirement dinner, then exactly how many different groups of four are there each of which would be an acceptable selection?
The correct option is C Three
Since the second condition states "either E or F must be selected...," we can infer from the new supposition (F is not selected) and the second condition (either E or F, but not both, is selected) that E is selected. And since E is selected, we know from the third condition that C is selected. In other words every acceptable selection must include both C and E.
We are now in a good position to enumerate the groups of four which would be acceptable selections. The first condition specifies that either A or B, but not both, must be selected. So you need to consider the case where A (but not B) is selected and the case in which B (but not A) is selected. Let's first consider the case where A (but not B) is selected. In this case, G is not selected, since the fourth condition tells you that if B is not selected, then G cannot be selected either. Since exactly four people must be selected, and since F, B, and G are not selected, D, the only remaining person, must be selected. Since D's selection does not violate any of the conditions or the new supposition, E, C, A, and D is an acceptable selection; in fact, it is the only acceptable selection when B is not selected. So far we have one acceptable selection, but we must now examine what holds in the case where B is selected.
Suppose that B is selected. In this case A is not selected, but G may or may not be selected. If G is selected, it is part of an acceptable selection -- E, C, B, and G. If G is not selected, remembering that A and F are also not selected, D must be selected. This gives us our final acceptable selection -- E, C, B, and D.
Thus there are exactly three different groups of four which make up acceptable selections, and (C) is the correct option.
Three
Since the second condition states "either E or F must be selected...," we can infer from the new supposition (F is not selected) and the second condition (either E or F, but not both, is selected) that E is selected. And since E is selected, we know from the third condition that C is selected. In other words every acceptable selection must include both C and E.
We are now in a good position to enumerate the groups of four which would be acceptable selections. The first condition specifies that either A or B, but not both, must be selected. So you need to consider the case where A (but not B) is selected and the case in which B (but not A) is selected. Let's first consider the case where A (but not B) is selected. In this case, G is not selected, since the fourth condition tells you that if B is not selected, then G cannot be selected either. Since exactly four people must be selected, and since F, B, and G are not selected, D, the only remaining person, must be selected. Since D's selection does not violate any of the conditions or the new supposition, E, C, A, and D is an acceptable selection; in fact, it is the only acceptable selection when B is not selected. So far we have one acceptable selection, but we must now examine what holds in the case where B is selected.
Suppose that B is selected. In this case A is not selected, but G may or may not be selected. If G is selected, it is part of an acceptable selection -- E, C, B, and G. If G is not selected, remembering that A and F are also not selected, D must be selected. This gives us our final acceptable selection -- E, C, B, and D.
Thus there are exactly three different groups of four which make up acceptable selections, and (C) is the correct option.
