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With no restrictions, then for each of the seven people, we'd have 2 choices: the person goes on the trip or does not. So with no restrictions, the answer would be 2^7 = 128.

Now among those 128 groups, in 1/4 of them neither R nor H go, in 1/4 of them R goes and H does not, in 1/4 of them R does not go and H does, and in 1/4 of them R and H both go. So we only want to count 3/4 of these groups, and the answer is (3/4)(2^7) = 96.

Or you could think about it this way: with no restrictions we have 2^7 groups. If R and H did go together, we then have two choices for each of the 5 remaining people, for 2^5 groups. We don't want to include those 2^5 groups, so the answer is 2^7 - 2^5 = 2^5(2^2 - 1) = 32*3 = 96.
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With no restrictions, then for each of the seven people, we'd have 2 choices: the person goes on the trip or does not. So with no restrictions, the answer would be 2^7 = 128.

Now among those 128 groups, in 1/4 of them neither R nor H go, in 1/4 of them R goes and H does not, in 1/4 of them R does not go and H does, and in 1/4 of them R and H both go. So we only want to count 3/4 of these groups, and the answer is (3/4)(2^7) = 96.

Or you could think about it this way: with no restrictions we have 2^7 groups. If R and H did go together, we then have two choices for each of the 5 remaining people, for 2^5 groups. We don't want to include those 2^5 groups, so the answer is 2^7 - 2^5 = 2^5(2^2 - 1) = 32*3 = 96.

Yes, 2^7 - 2^5 was the way I solved it.

P.S. This is my question. Hope the wording is precise.
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Essentially, rather than making chooses and selections for each scenario, we can decide from each individual person’s perspective whether they will accompany Harry.

(1st) Total Number of Ways Harry can go assuming there were NO CONSTRAINTS

Harry must go: 1 available option

And

Ron: can either go or not go: 2 available options

And

Etc.

For each person, they can either go or not go on the trip and join Harry - 2 available options


Total ways Harry can travel assuming no constraints = (1) * (2)^7


(2nd) Subtract our the Unfavorable Groupings in with Ron and Hermoine are together with Harry

Both Ron and Hermoine are chosen to go with Harry, then for each of the remaining people they can either go or not go:

Hagrid: can either join or not join the couple - 2 available options

And

Luna: can either join or not join - 2 available options

Etc.
For the remaining 5 people

Number of unfavorable groupings in which Ron and Hermoine are together on the trip with Harry = (1) * (1) * (2)^5


Answer:

(2)^7 - (2)^5 =

128 - 32 =

96

Posted from my mobile device
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Given: Harry is planning a journey to Hogwarts. He can go alone or with any number of his 7 friends: Ron, Hermione, Hagrid, Luna, Neville, Fred and George.

Asked: If Ron and Hermione refuse to go together, how many groups are possible for the journey ?

Total number of ways Harry’s friends may join him = 2^7
Number of ways in which Ron and Hermione are together = 2^5
Number of possible groups for the journey = 2^7- 2^5 = 2^5 (2^2-1) = 32 *3 = 96

IMO C
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