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28 Jul 2021, 02:21
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C
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Difficulty:
85%
(hard)
Question Stats:
55% (02:43) correct
45%
(03:14)
wrong
based on 201
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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
M37-20
A. 88
B. 95
C. 96
D. 1,560
E. 3,600
M37-20
Attachment:
Harry.png
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19 Nov 2021, 09:54
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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
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
General Discussion
Arjnath
Joined: 19 Oct 2020
Last visit: 28 Jan 2024
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Schools: Booth '24 Columbia CBS '24 Fuqua '24 Haas '24 Harvard '24 HKUST ISB '23 Kellogg '24 LBS '24 NUS Said '23 Erasmus Sloan '24 Stanford '24 Tepper '24 Wharton '24 Yale '24
Schools: Booth '24 Columbia CBS '24 Fuqua '24 Haas '24 Harvard '24 HKUST ISB '23 Kellogg '24 LBS '24 NUS Said '23 Erasmus Sloan '24 Stanford '24 Tepper '24 Wharton '24 Yale '24
Posts: 3
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28 Jul 2021, 05:01
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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.
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.
