Harry is planning a journey to Hogwarts. He can go alone or with any

Official Solution:

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

A. \(88\)
B. \(95\)
C. \(96\)
D. \(1,560\)
E. \(3,600\)


How many groups are possible if we did not have the restriction? Without the restriction the total number of groups possible is \(2^7\): each of Harry's 7 friends can either join Harry or not.

How many groups are possible with Ron and Hermione in them? If Ron and Hermione are in the group, then each of the 5 remaining friends can either join or not, so the number of groups with Ron and Hermione is \(2^5\).

So, there are \(Total-Restriction=2^7-2^5=2^5(2^2-1)=96\) groups possible.


Answer: C
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First started with visualizing different sizes of the group. It can be 1 (H alone),2 (i.e.H+1),3 (H+2).....8(H+7)
Second - R and He can't go together.
Third -
Read as
Group size Total poss ways Excl of R&He

1 1
2 7c1
3 7c2 5c0
4 7c3 5c1
5 7c4 5c2
6 7c5 5c3
7 7c6 5c4
8 7c7 5c0


Let me start with Group size of 2 ie, H + any one of seven people so - 7c1 ways
Group size 3 ie. Any two of seven people.... If the two selected people are R&He, we dont want to select that group, that can be found out as 5c0. i.e. not selecting anyone of the other 5 people.. So the ways would be 7c2 - 5c0

Very similar arguments for the other sizes.. let me jump to Group size 6, here it is Harry + 5 more people... Similarly we can select 5 out of the seven people in 7c5 and we want to exclude the selection where only 3 out of the 5 people (i.e. except R&He) are selected. This indirectly means R&He are selected. The number of ways would be 7c5-5c3

After which it is straightforward addition and subtraction which would result in total ways to be 96.

The question tested whether one could visualize the selection of two, as a negation of not selecting others. This is my first post - so not sure if I covered all aspects required to communicate. pls let me know if I should explain somewhere better and hoping the answer is correct.

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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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

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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