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GMAT Club Olympics 2025 (Day 6): An old photograph shows a group of

1 2 3 4 5

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Bunuel

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Total number of people = N
4 out of 10 people from the left are wearing hats and all remaining people in the row are wearing hats.
Eg: From 1 to 10 : 4 wear hats and 6 don't.
From 11 to N: All wear hats.
N= ?
For DS we only have to check whether the statement would help us find a solution or not.

Statement 1.
5 of 6 people from the right are wearing hats.
So, we have the information from the right.
We have the people from right and the people from left.
Hence, we can find out the number of people in the row.
Sufficient.

Statement 2:
3 of 8 people from the left are wearing hats.
This statement becomes redundant and we cannot find the number of people in the row.
Not Sufficient.

Answer: A

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ANSWER IS A

Explanation:

The question gives us the number of people on the LHS which is 4 out of 10 people where hats - we just need to know the number of people on the RHS.

1) Give us -number of people on RHS-which is exactly that we need - SUFFIECIENT
2) Gives us people on the LHS. We have no idea about RHS-NOT SUFFICIENT

HENCE A.

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08 Jul 2025, 01:15

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let us assume that there are x people in the row to make the reasoning simple

we know that 4 of 10 people from left are wearing hats. and all others are wearing hats. so only 6 are not wearing hats in total

Statement 1: 5 of 6 people from right are wearing hats.
so 1 is not wearing hat from right,

combining both statements we get 11 people as then only the equation is satisfied. if we increase then number of people not wearing hats will increase. if we decrease number of people not wearing hats will decrease

statement 2: 3 of 8 people from left are wearing hats
so we know that one person on the position 9 or 10 from left is wearing hat

but not useful

So only statement 1 is enough to answer, statement 2 is insufficient.

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Correct Answer: There must be at least 15 person in the row and this can be find out using statement I alone.
Given we have,
From 1-10 person, 4 person are wearing hat

Now let's take Statement I,

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

If we take total as 15,

In this case from position 10-15 at least 5 people are wearing hats.
So from first 10 position and last 6 position, position share is only one person. This position is 10th.
So total must be greater than 15

If we take total as 16,

In this case 1-10 position have 4 person wearing hat and from 11-16, 5 person was wearing hat. In this case no position is shared by a person.
But as per the statement, from last 6 person, 1 person is not wearing hat, this makes wrong because as per given information from 1-10 position 4 are wearing hat and rest others are wearing hat. So this case is incorrect.

Now let's take Statement II,
3 of the 8 people from the left are wearing hats.

If we take only this, we will get multiple cases and also we cannot find exact number of members.

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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?

(1) 5 of the 6 people from the right are wearing hats.
Total on the left, we know 4 are wearing hats, and we do not know the positions.
If we take the worst case scenario, N=15, then we can say the last 5 are wearing hats and 4 from the first 10, but if we take N=14, then also this works.
Insufficient
(2) 3 of the 8 people from the left are wearing hats- this doesn't give us any information.
Insufficient
Combining these would again not give us any new information so E
IMO:E

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E. Statements (1) and (2) TOGETHER are NOT sufficient.

Let's assume the image below is the group of people
10 x
<----------><------->
Out of the 10 people, we know 4 are wearing hats. We also know all of the 'x' people are wearing hats too.

(1) 5 of the 6 people from the right are wearing hats
We know that the x people standing on the right are wearing hats, from the first statement we can now tell that x has to be less than or equal to 5 because they all have to be wearing hats. Not sufficient, since answer can be anywhere between 11 to 15

(2) 3 of the 8 people from the left are wearing hats
This implies that either the 9th or 10th person is wearing a hat, but we still don't know anything about 'x'.

Combining these two,
Let's say, if the 9th person is wearing a hat, we have two cases.

i) / / / / / / / / | _ | | | |

ii) / / / / / / / / | _ | | | | |

('/' represent people who might or might not be wearing hats, '|' represent people who are wearing hats, '_' represent people who are not wearing hats)
Out of the first 8 '/' any 3 can be wearing hats and all the conditions will be satisfied but we still don't have a specific answer,
hence, E

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08 Jul 2025, 02:43

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Bunuel

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4 out of first 10 people wear hats. Let's denote Hats by h and no-Hat by '0'
Rest everyone wears a hat hence from person 11 onwards everyone wears a hat

(1) 5 of the 6 people from the right are wearing hats.
Let's try some arrangements
hhhh000000hhhhh
or
hhhh000000000000hhhhh
or
000000hhhhh

Multiple configurations are possible

Insufficient

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

Let's try some arrangements
hhh00000h0hhhhh
hhh000h000000hhhhhhhh

Multiple configurations are possible

Insufficient

Combining both together we get

5 of the 6 people from the right are wearing hats.
and
3 of the 8 people from the left are wearing hats.

hhh00000h0hhhhh
or
h0h0h000h0hhhhh

Any 3 of the first 8 can wear hats.
The ninth person has to wear the hat and the 10th person cannot.
Following 11 to 15 will wear hats

In any way the number of people is restricted to 15

Sufficient

Option C

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Bunuel

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|--------------10 people (4 hats)--------------------|----------------all hats ---------------|

1. all 5 wearing hats has to be the extreme right

|--------------9 people (4 hats)------|---1 no hat--|-----------------5 hats ---------------|

There are 15 people with 9 hats. - Sufficient.

2.

|--------------8 people (3 hats)----|-----2 people (1 hat)-----|----------------all hats ---------------|

We have no idea, how many people are there in the right who are wearing hats ---Insufficient.

Hence A is the answer.

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08 Jul 2025, 04:22

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here's what we are given : In the first 10 people from the left, exactly 4 are wearing hats. So the other 6 in that group are not. Every person beyond the 10th is wearing a hat.

Statement 1 : 5 of the 6 people from the right are wearing hats.
It basically tells us that in the 6 people on the far right, only 1 is without a hat. We also know that the only people without hats are in the first 10, so the right-hand group of 6 must have an overlap with the 1st group.
Then the total no. of people can be anywhere between 11 to 15. Insufficient.

Statement 2 : 3 of the 8 people from the left are wearing hats.
This tells us that among the first 8 people, 3 have hats, so 5 do not. But this info only tells us about another arrangement of initial info and tells us nothing about people beyond the 10th position. Insufficient

Even if we take both statements together, total no. of people can still be anywhere between 11 and 15. Insufficient

Answer : E

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(1)
Examples:
11 people, people with hats: 6,7,8,9,11
12 people, people with hats: 6,7,8,9,11,12

Statement is insufficient

(2)
Examples:
11 people, people with hats: 6,7,8,9,11
12 people, people with hats: 6,7,8,9,11,12

Statement is insufficient

(1)+(2)
Examples:
11 people, people with hats: 6,7,8,9,11
12 people, people with hats: 6,7,8,9,11,12

Both statements are insufficient

The right answer is E

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08 Jul 2025, 04:46

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Given
From left - 4 out of 10 are wearing hats - this can be in any sequence - as long as any 4 are wearing hats when taken 10 from left

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

Lets take a case of N=11

Here there are 11 people, we can see that this fulfil all constraints
>> 4/10 from left are wearing hats, remaining 1 person to the right of the 10th - which is the 11th - is also wearing hat
>> 5/6 from right are wearing hats

But the case of N= 12 is also possible :

Here we have 12 people but this also fulfils all constraints
>> 4/10 from left wearing hats, remaining 2 people to the right of 10th person - 11th and 12th - are also wearing hats
>> 5/6 from right are wearing hats

So we have 2 possible cases and we do not have a fixed number of people that would fulfil these constraints. Therefore statement 1 is not sufficient.

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

This tells us that 1 person from 9 or 10th position is wearing hat, but does not give us more information on total number of people. There could be 5 people to the right of the 10th person, or 10 people - we dont know.
Statement 2 is insufficient

(1) and (2) together

Let us again take the 2 cases we took earlier

N = 11, with some rearranging:

>> 4/10 from left are wearing hats, remaining 11th position is also wearing hat.
>> 3/8 from left are wearing hats
>> 5/6 from right are also wearing hats

N = 12, after some rearranging :

>> 4/10 from left are wearing hats, remaining 11th & 12th position is also wearing hat.
>> 3/8 from left are wearing hats
>> 5/6 from right are also wearing hats

So again we have 2 cases N =12 and N=11 possible. Statement 1 and 2 together are also not sufficient

Answer is E

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Option A is the correct answer.

Lets understand the information mentioned in the question and what we need to find in order answer the question.

The question starts by telling us that their is a group of people in a photograph who are standing in a row. Among all the people in the photo 4 of the first 10 people from the lest are wearing hats and for the rest n people they all are also wearing the hats. Now the question asks us how many people are there in the photo. Lets use diagrams for our ease.

The diagram represent the 10 people from the left standing in the row. In this the colored circle represent the people wearing the hats.

Now lets check the given statements and see if we can answer the question using them or not.

Statement 1: "5 of the 6 people from the right are wearing hats". This statement tells us the 5 out of first 6 people from the right are wearing a hats. Now if we are not careful with the information mentioned in the question and mention in this statement then we will surely make a calculation mistake i.e. we will assume that from the left information for 10 people are given and from the right information for 6 are given so there could or could not be people between these two groups which will be the wrong assumption because the question clearly tells us that "4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats" which would mean that their is a overlap in this set i.e. the 1 person who is not wearing the hat in the group of 6 people from the right also belong to the group of 10 people from the left. Lets see it in the diagram for better clarity. So from the below diagram we can confidently say that we will get the confirmed answer from this statement which will will be: 10 + 6 - 1 => 15 people in the Row. So Statement 1 is Sufficient.

Statement 2: "3 of the 8 people from the left are wearing hats". This statements just tells us the information about the people from the side side of the row which we already know the the question as we can see in the image below as well. And from here we can assume the people from the right side could be any number like: 0, 1, 2, 10 and so on so this statement is Not Sufficient to answer.

So here we can conclude that Only first Statement alone is sufficient to answer the question that's why Option A is our answer.

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08 Jul 2025, 05:50

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E, there is no info leading up to the conclusion of actual how many people are in the row, though they do provide about how many people are wearing hats from left and right, couldnt find any way to calculate the asked.

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Given -> 4 of the 10 people from the left wear hats and the remaining people in the row also wear hats.
To determine -> Number of people in the row

Statement 1 -> 5 of the 6 people from the right are wearing hats.
If 5 of the 6 people from the right wear hats, then 1 person in these 6 people from the right does not wear a hat. But the only people who don't wear hats are in the leftmost 10. That means there is an overlap.
It is possible that the first 4 people from the left are hat wearers and the remaining 6 are non-hat wearers. That would mean the 1 person who doesn't wear a hat in the rightmost 6 is in the 10th position. After the 10th position, there are 5 more people who wear hats. Therefore, total number of people in this row are 5.
It is also possible that the first 6 people from the left are non-hat wearers. That means the next 4 people are hat wearers. Now, from the 6 rightmost people, 1 of them does not wear a hat. This person has to be in position 6. That means the 4 people who wear a hat in the 10 leftmost are counted in the 5 people who wear a hat in the 6 rightmost. So, there is 1 extra person who wears a hat. Therefore, total number of people in the row are 11.
We got 2 different answers from statement 1. Not sufficient.

Statement 2 -> 3 of the 8 people from the left are wearing hats.
This tells us out of the 10 leftmost people, the first 8 have 3 hat wearers. That means people in position 9 and 10 are either hat-nonhat or nonhat-hat respectively. But this statement doesn't tell us anything else. insufficient.

1+2 ->
If 9th and 10th person from the left are hat-nonhat, then this nonhat person is included in the 6 rightmost people. Then the 5 rightmost people are hat wearers. Then total people = 10
+ 5 = 15.
If 9th and 10th person from the left are nonhat-hat, then both these people are included in the 6 rightmost people. Then the 4 rightmost people are hat wearers. Then total people = 10
+ 4 = 14.
We got 2 different answers by combining both the statements. Both statements together are not sufficient.

Answer - E

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08 Jul 2025, 06:29

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Ans (A)

we are told 4/10 from left are with hats and the rest are also hats.

Statement 1 5/6 from right have hats

this means that the peope from left are overlapping with the people fron the right since as per question stem we are given that 4/10 have hats and the rest in the row also have hats. This means that 4 hats+ 6 no hats + 5 hats. there are 15 people in the row

Statement 2 3/8 from the right have hats

Not sufficient since it does not tell us anything about the people beyond the 10

Bunuel

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08 Jul 2025, 06:41

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Given,
In a photograph, a group of people are standing in one row.

Let’s denote people with hats with tick mark ✓
And people without hats with x

Since 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats, we can assume n=11 where

Any 4 people could be wearing hats and all other people starting from 11[sup]th[/sup] are wearing hats

✓xxx✓xx✓x✓✓

Let’s analyze the statements to know how many people are there in total other than these 10 people.

St1: 5 of the 6 people from the right are wearing hats.

Let’s take the above scenario where n=11

xxxxx✓✓✓✓x✓

Total 4 out of 11 are wearing hats

When n=14

xxxxx✓✓✓x✓✓✓✓✓

Both conditions are met and Total 7 out of 14 are wearing hats.

We already have 02 cases possible. So, St1 is not sufficient.

St2: 3 of the 8 people from the left are wearing hats.

Conditions:
i. 4 people out of 10 from the left are wearing hats
ii. 3 people out of 8 from the left are wearing hats

When n=11

xx✓✓✓xxx✓x✓ which gives: Total 5 out of 11 are wearing hats.

We don’t know how many people beyond these 10 people are wearing/not wearing hats. Insufficient.

Combing St1 and St2:

Conditions:
i. 4 people out of 10 from the left are wearing hats
ii. 5 of the 6 people from the right are wearing hats.
iii. 3 people out of 8 from the left are wearing hats

All the conditions could be met for scenarios n=12 and when n=14

Together also, we are unable to find unique value of n.

Option E

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08 Jul 2025, 06:45

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S1 If 5/6 from the right have hats and the question states that from the 11th person have hats then it means those 5 of the six must be the rest of the people all of whom have hats meaning the total number of people is 10+5 = 15 thus sufficient
S2 Insufficient because either no 9 or 10 could have a hat and may or may not be part of the rest all of whom have hats
Hence A

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08 Jul 2025, 07:04

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(1)
Two or more possibilities:
14 people are possible if hats are in the positions: 4,5,6,9,11,12,13,14
15 people are possible if hats are in the positions: 4,5,6,9,11,12,13,14,15

Insufficient

(2)
Two or more possibilities:
14 people are possible if hats are in the positions: 4,5,6,9,11,12,13,14
15 people are possible if hats are in the positions: 4,5,6,9,11,12,13,14,15

Insufficient

(1) and (2)
Two or more possibilities:
14 people are possible if hats are in the positions: 4,5,6,9,11,12,13,14
15 people are possible if hats are in the positions: 4,5,6,9,11,12,13,14,15

Insufficient

Correct answer is E

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08 Jul 2025, 07:32

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The easiest way to solve it is to find possible solutions.

(1)
There are more than one solution:
13 people, hats in 2, 3, 8, 9, 11, 12, 13.
14 people, hats in 2, 3, 8, 9, 11, 12, 13, 14.

Statement (1) alone is insufficient.

(2)
There are more than one solution:
13 people, hats in 2, 3, 8, 9, 11, 12, 13.
14 people, hats in 2, 3, 8, 9, 11, 12, 13, 14.

Statement (2) alone is insufficient.

(1)+(2)
There are still more than one solution:
13 people, hats in 2, 3, 8, 9, 11, 12, 13.
14 people, hats in 2, 3, 8, 9, 11, 12, 13, 14.

Statement (1) and (2) together are insufficient

Answer E

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08 Jul 2025, 07:41

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IMO, correct answer is A.

Its is given in the question that , 4 of the 10 people from the left are wearing hats and all remaining people in the row are wearing hats.

Statement I:
Based on the above information, we can definitively say that all people after 10th person from left are wearing hats. If 5 out of 6 people from right are wearing hats, that 1 person , who is not wearing hat must be the same as the 10th person from the left. So, in total, there are (10 + 6 -1) = 15 people in the row. Sufficient.

Statement II:
3 of the 8 people from the left wearing hats. It is just a subset of what is already provided in question stem. Therefore, insufficient.

Bunuel

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

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