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07 Jul 2025, 12:56
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Bunuel
(1) 5 of the 6 people from the right are wearing hats.
The question just mentions that 4 of the 10 people from the left are wearing hats but the number of people are not mentioned. Hence, between the 10th person from left and the 6th person from right we can have zero to any number of people. Also, the members can overlap.
Statement 1 doesn't help evaluate the number of people in the row.
(2) 3 of the 8 people from the left are wearing hats.
If 4 out of 10 people from the left are wearing hats we know that 3 of the 8 people from the left are wearing hats.
Combined we can't find the answer as the number of people between two points is not known.
Option E
MBAChaser123
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07 Jul 2025, 13:45
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Statement 1:
If 5 of the 6 people from the right are wearing hats, we have at least 1 person more than 10, since the first 10 people have to have only 4 hats. But, we can also have 5 extra people over 10, since the one without a hat has to be among the first 10, to satisfy the rest of the question stem, which tells us the remaining people on the right all have hats.
There are also options in between these, so statement 1 is not sufficient.
Statement 2:
This statement tells us that of the last 2 people of the first 10, only 1 has a hat. This statement is also not enough. There is no limit to the number of people on the right of the 10 people, as long as they all have hats, we have infinite options.
Both statements together:
If we consider both statements together, depending on which of the last two from the first ten has hat, we have a different answer,
So both statements together are not sufficient.
The answer is E.
If 5 of the 6 people from the right are wearing hats, we have at least 1 person more than 10, since the first 10 people have to have only 4 hats. But, we can also have 5 extra people over 10, since the one without a hat has to be among the first 10, to satisfy the rest of the question stem, which tells us the remaining people on the right all have hats.
There are also options in between these, so statement 1 is not sufficient.
| The first ten | The rest | |
| Minimum possible | i i i i i i T T T T | T |
| Maximum possible | i T i T i T i T i i | T T T T T |
Statement 2:
This statement tells us that of the last 2 people of the first 10, only 1 has a hat. This statement is also not enough. There is no limit to the number of people on the right of the 10 people, as long as they all have hats, we have infinite options.
Both statements together:
If we consider both statements together, depending on which of the last two from the first ten has hat, we have a different answer,
So both statements together are not sufficient.
| The first eight | 9th and 10th | The rest | |
| First option | T T T i i i i i | i T | T T T T |
| Second option | T T T i i i i i | T i | T T T T T |
The answer is E.
Bunuel
07 Jul 2025, 14:17
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Row: |______Y_______|_____X____|
Y = Size 10. 4 people are wearing hats.
X = Unknow size. Everyone is wearing hats.
Rephrasing the question: X + Y = ?
(1) 5 of the 6 people from the right are wearing hats.
Since we know that every person after the 10-th position is wearing a hat, we can infer that there's a overlap between the "6 people from the right" and the "10 people from the left". Is this information sufficient? No, it's not. The size can be from 11 (since we havce 4 people with hat in 10 first positions, we need at least one more person with hat) to 15 (since everyone after the 10-th position must wear a hat, 5 from this 6 people can be after 10-th position).
Eliminate answer choices A and D.
(2) 3 of the 8 people from the left are wearing hats.
Ok. So we now know that, in positions 9 and 10 we have exactly 1 person wearing a hat and 1 person no wearing a hat. Is it sufficient? No, it's not. We can have know infinite options, since we do not have any bound to the right after 10-th position.
Eliminate answer choice C.
Statements (1) and (2)
Plugging both information we can have:
Row: |___8____|_2_|___X___|
In the 8 first positions, 3 hats. Between positions 9 and 10 we have one more hat.
If we have H in position 10, with statement (1) we can have 1 option:
<8 initial positions>NH HHHH. Size = 8 + 6 = 14. Where H represent a person wearing hat and N represent a person not wearing a hat.
If we have H in position 9, with statement (1) we can have 2 options:
<8 initial positions>HN HHHH. Size = 8 + 6 = 14.
<8 initial positions>HN HHHHH. Size = 8 + 7 = 15.
Therefore, we are not able to determine the row length only with the information provided. Eliminate answer choice C.
Answer = E.
Y = Size 10. 4 people are wearing hats.
X = Unknow size. Everyone is wearing hats.
Rephrasing the question: X + Y = ?
(1) 5 of the 6 people from the right are wearing hats.
Since we know that every person after the 10-th position is wearing a hat, we can infer that there's a overlap between the "6 people from the right" and the "10 people from the left". Is this information sufficient? No, it's not. The size can be from 11 (since we havce 4 people with hat in 10 first positions, we need at least one more person with hat) to 15 (since everyone after the 10-th position must wear a hat, 5 from this 6 people can be after 10-th position).
Eliminate answer choices A and D.
(2) 3 of the 8 people from the left are wearing hats.
Ok. So we now know that, in positions 9 and 10 we have exactly 1 person wearing a hat and 1 person no wearing a hat. Is it sufficient? No, it's not. We can have know infinite options, since we do not have any bound to the right after 10-th position.
Eliminate answer choice C.
Statements (1) and (2)
Plugging both information we can have:
Row: |___8____|_2_|___X___|
In the 8 first positions, 3 hats. Between positions 9 and 10 we have one more hat.
If we have H in position 10, with statement (1) we can have 1 option:
<8 initial positions>NH HHHH. Size = 8 + 6 = 14. Where H represent a person wearing hat and N represent a person not wearing a hat.
If we have H in position 9, with statement (1) we can have 2 options:
<8 initial positions>HN HHHH. Size = 8 + 6 = 14.
<8 initial positions>HN HHHHH. Size = 8 + 7 = 15.
Therefore, we are not able to determine the row length only with the information provided. Eliminate answer choice C.
Answer = E.
