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Total number of people: 10 + x
It is given that 4 from left out of 10 are wearing hats.
It is given that all remaining are wearing hats.
|-----------10-----------|-----x----|

Statement 1:
5 out of 6 from right are wearing hats. This means that x could be 1 and the last 4 of the 10 from the left are wearing hats. OR x could be 4 and the last one among the 10 from left could be wearing a hat. NOT SUFFICIENT.

Statement 2:
3 of 8 from left are wearing hats. NOT SUFFICIENT.

Combining:
4 out of 10 and 3 out of 8 statement is telling us that 10th from left has a hat and the 9th one does not have a hat.
5 out of 6 from right are wearing hats which means x = 4.

Answer is C.
Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


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Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


This question was provided by GMAT Club
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1) Alone not sufficient. We are given information about the left start of the row. We know (Roughly): 6xH and 4xN (No hat). But we dont know, if the last hat is positioned on the 10th, or the 6th place.
2) Not sufficient alone, as this only hints, that the last person with a hat must be at position 9 or 10.

Together they are not sufficient as well, as it is still open, if the position at the 9th, or the 10th place is held by a person with a hat.

As a result, the answer is E)
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Topic(s)- Overlapping Sets(, Permutations)
Strategy- Visualization
Variable(s)- total from the left = "L"; total remaining people not from the left = "nL"; total from the right = "R"; total overlap from the left and from the right = "LR"

0. Pre-Work: Visualization
From the left Remaining People
4 hats/10 total (nL) hats/ (nL) total = 1/1

Rephrase the Question: What is nL?

(1) R = 5/6 hats
Since ALL of the remaining in the row wear hats, this implies there is at most 1 person "from the left" and "from the right"
i. From the left From the right
4 hats/10 total 5 hats/6 total
ii. From the left Remaining People
4 hats / (10+1=11 total) 5 hats / (6-1=5 total)
nL = 5 [Eliminate B,C,E]

(2) This is a subset of the "original" from the left group
nL? [Eliminate D]

Answer: A
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Given:
1)In the first 10 people from the left ,4 are wearing hats.
2)All remaining people in the row are also wearing hats

Let total people wearing hats = x
So total wearing hats = (x-10)+4 = (x-6)

Statement(I):
1)Lets test x=14,
hats= 14-6= 8
Last 6 people= position 9 to 14
position 11-14 - all hats(from question)= 4 hats
Need 1 more hat from position 9 or 10..possible

2)x=15
hats = 15-6=9
Last 6 people= position 10 to 15
Position 10 may or may not wear hat, remaining 5 can all wear hats...possible

Multiple values of x satisfy statement(I),so statement(1) alone not sufficient

Statement(II):
4 of the first 10 wear hats
So position 9 and 10 together must have 1 hat
This is possible for multiple values of x, including x=14, x=15

Statement(II) alone also not sufficient

Combining together we are getting 2 values of x=14,15, so not sufficient

Option E
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The correct answer is answer choice (D) Each statement alone is sufficient

Given the information in the question stem, we know that our goal is to figure out how many total people are in the row. We know that there are 10 people on the left side of the row as it is mentioned.

Statement 1 - this gives us the number of 6 on the right side of the row, so we can determine the 10 on the left from the question stem + 6 from this statement would be 16 total. This statement is sufficient.

Statement 2 - this gives us the number of 8 on the right side of the row, so we can determine the 10 on the left from the question stem + 8 from this statement would be 18 total. This statement is sufficient.

The answer choice that aligns with both statements being independently sufficient is D.
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Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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See, we know from left 4 out of 10 are wearing hats. So

From statement 1, we can understand 5 out of 6 from right wear hat too which means 1 of them from right is not wearing hat. while rest still wear hats. But in question it was said that all in right after 10 people wear hat. So it means the 1 person in right is actually in those 10 people from left (basically the rows from right and left merge at some point and a scenario of double calculation is happening. Which means total number of people is less than or equal to 15. (best case being when the 10th people doesnt wear hat so that case we get 15, else could be lower. But it tells nothing about number clearly so not sufficient.

Now from second equation, we can understand that 3 out of 8 people from left wear hat too, but it tells us nothing how to find total so this is not sufficient too.

But when we consider both equation together, we get to know that 3 out of 8 in left wear hat which means 1 of the 10 left people is after 8th person ... correct... so it is either 9th or 10th. Now, if the 10th person wear hat then the merge will never happen... because it would make the right most people have one person who does not wear hat which contradicts statement in question so it must have 9th person wearing hat and 10th person not wearing. Which means their is merge of 1 person happening and total number of people will be 15. So both equation together is sufficient but alone does not. (C)
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An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

T = 4H + 6NT + OH; 4 of the 10 people are wearing hats and the other wearing hats. This suggests that this was observed from somewhere in between, not from the extreme left or right.

(1) 5 of the 6 people from the right are wearing hats.

XXXXXXHHHHXH; XXXXXHXHHHHH; From the given data, these two combinations are possible when observed from in between.

Hence, Total = 12; Sufficient.

(2) 3 of the 8 people from the left are wearing hats.

XXXXXHHH;XXXHHHXX; Like this, there could be many combinations, but none of them give us anything about the total nos. Hence Insufficient.

Ans A
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A. (1) by itself

(1)
NNNNNHHHNHHH
The first H can change position to the left, last N (no hat) can change position to max 6th position (right to left). But you can't add/take, it's 12 fixed. SUFFICIENT

(2)
You can add H to the right indefinitly. NOT SUFFICIENT
Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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We know that the left 10 have 4 hats. We need to find the total people.

Stmt 1 : 5 of the 6 people from the right are wearing hats.
If n=11,

1 - 10, The right most contains 4 Hats and 1 Non Hat, 1 on the right most of the 10 is also Hat. Hence the 6 right most from the 11, 5 Hats.

Even if n=12,
1-10, Right 2 hatss and Of the 10 3 Hats and 1 Non Hat. So n can be from 11 to 15 all possible.

Hence insufficient.

Stmt 2 : 3 of the 8 people from the left are wearing hats.

The right of 10 can be any number, So this is not sufficient.


Using both the statements we can have n as 11 or 12.

Hence IMO E
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(1)
There must be at least 11 people (the fifth hat will be in 11) and no more than 15 (if there were 16 then 6 of 6 people were wearing hats).

INSUFFICIENT

(2)
We cannot deduce how many people are in the right of the first 10 people.

INSUFFICIENT

(1)+(2)
11 people are possible if hats are, for example, in 6, 7, 8, 9 and 11.
12 people are possible if hats are, for example, in 1, 7, 8, 9, 11 and 12.
13 people are possible if hats are, for example, in 1, 2, 8, 9, 11, 12 and 13.
14 people are possible if hats are, for example, in 1, 2, 3, 9, 11, 12, 13 and 14.
15 people are possible if hats are, for example, in 1, 2, 3, 9, 11, 12, 13, 14 and 15.

INSUFFICIENT

IMO E
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Statement (1)

We are not sure where the 1 non-hat wearing person is seated. A and D are out.


Statement (2)

While we know that the first 10 people from left 4 wore hats, this statement doesn't give any information on how many are on the right. So B is out.


Statement (1) and (2)

Borrowing 1 person from the right to fulfill 4th hat person on the left, we can successfully arrive at a conclusive number. C is right.


Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

 

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Statement (1):
The last 6 people include 5 hat wearers, but the problem states all people beyond the first 10 wear hats. This can only happen if the total number of people is exactly 16 (so the right segment is 6 people) and one person in that segment does not wear a hat, contradicting the “all wear hats” unless the row ends there.
Thus, Statement (1) is sufficient to find N = 16.

Statement (2):
Knowing that 3 of the first 8 wear hats refines how hats are distributed among the first 10 but tells us nothing about how many people are in the entire row or beyond the 10th person. This leaves total N unknown, so Statement (2) alone is insufficient.

Therefore, Statement (1) alone is sufficient.
Answer: A.
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An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

Statement 1
5 of the 6 people from the right are wearing hats.

Suppose Total no of people = N

five out of Six from right are wearing hats and four out of 10 from left are wearing hats

So N Can be 13, 14, 15 ,16, etc
Multiple answer can be there.
So insufficient.

Statement 2
3 of the 8 people from the left are wearing hats.
from this any one of 9 & 10 can have hats

we have no information how many are in right side

So statement is insufficient.

Lets combine statements

out of 9 and 10 one isn't wearing hat
lets say 10 isn't wearing hat

so all after that will wear hats
and total 5 out of 6 are wearing hats from right
Lets say N= 14
11, 12, 13 & 14 wear hats
9 wear hats, 1 to 8 has 3 who wear hats

lets say 10 wears hat and 9 has no hat
again last five from right wears hat
4 of first 10 have hats. 1to 8 has 3 hats
all statements are met.

So answer is C as only one unique answer is followed i.e. N=14
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An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

H-> Person wearing hat
N-> Person not wearing hat

Let x number of people from the right are wearing hats
(4H6N)(xH)
Total number of people in the row = 10 + x

(1) 5 of the 6 people from the right are wearing hats.
x = 5
(4H5N (N) 5H)
The number of people in the row = 10 + 5 = 15
SUFFICIENT

(2) 3 of the 8 people from the left are wearing hats.
(3H5N NH)(xH)
With the information, the value of x can not be ascertained, therefore, the number of people in the row can not be derived.
NOT SUFFICIENT

IMO A
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An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

Statement 1: 5 of the 6 people from the right are wearing hats.

Since all people after 10th from left are wearing hats, from the right side, the first person to not to wear hat will be in the 10 people from left

So there are 5 from the right people are wearing hat. 4 in 10 people from left are wearing hat.

Totally \(10+5=15\) people are standing in a row

Sufficient

Statement 2: 3 of the 8 people from the left are wearing hats.


This is a subset of the given 4 in 10 people wearing hats.

This does not give any info of the remaining people.

Not Sufficient

Answer: A
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This cant be answered using both statements as-
4 out of 10 people from left are wearing hats.
Now all people starting form 11th will be wearing hats so according to 1st statement, from right 5/6 are wearing hats. But we dont know that is it overlapping with the 10 earlier mentioned. Similar is the case with 2nd statement.
So option E
Bunuel
An old photograph shows a group of people standing in a row. If 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, how many people are in the row?

(1) 5 of the 6 people from the right are wearing hats.
(2) 3 of the 8 people from the left are wearing hats.


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $30,000 in prizes such as Courses, Tests, Private Tutoring, and more

 

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Given: 4 of the first 10 people wear hats, everyone after position 10 wears hats.
(1) 5 of the last 6 people wear hats

  • This means 1 of the last 6 people doesn't wear a hat
  • But everyone after position 10 must wear hats
  • So that person without a hat must be in positions 1-10
  • This constrains the total to exactly 10 people
e.g If the total number of people is 15
    Then people 11 to 15 all wear hats. The last 6 people = people 10 to 15 Only person 10 might not wear a hat (since only 4 of the first 10 do). That fits — 5 of the last 6 wear hats
  • Try any number other than 15 — it won’t work.
    So we can find the total = 15.
  • Sufficient.
(2) 3 of the first 8 people wear hats
  • Given says 4 of first 10 wear hats
  • If only 3 of first 8 wear hats, then the 4th hat-wearer must be in positions 9-10
  • This is just more detail about the first 10 people, This doesn't tell us the total number of people
  • Insufficient
Answer: A
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