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08 Jul 2025, 07:47
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We know that,
4 of the first 10 people from the left are wearing hats
All the remaining people in the row are also wearing hats
Let the total number of people be T
Positions 1 through 10, 4 people wear hats
Positions 11 through T, all the people wear hats
Given statements,
(1) 5 of the 6 people from the right are wearing hats.
We know that, veryone from position 11 onward wears a hat, meaning only the first 10 may include non-hat wearers and now we also know that the last 6 people (from the right) include 5 wearing hats and 1 not wearing a hat
=> The last 6 people (positions T−5 to T) include some of the first 10 people (positions 1–10)
Now we want one of the last 6 people to be in the first 10
=> T- 5 <= 10
=> T <= 15
But we also want that, ONLY 1 of the last 6 people to be from position 1 - 10
=> T - 5 = 10
=> T = 15
So, lets try,
Positions 1–10 -> 4 hats, 6 no hats
Positions 11–15 -> all hats
For the last 6 people,
Position 10 = possibly not wearing a hat
Positions 11–15 = all hats which means 5 hats out of 6
So, T = 15 satisfies all the required conditions
Now,
If T < 15, more than one of the last 6 people would be in positions 1–10 which also contradicts the requirement
If T > 15, all of the last 6 people are beyond position 10 which contradicts the statement that only 5 of them wore hats.
So only possible value of T is 15.
(2) 3 of the 8 people from the left are wearing hats.
We know that, positions 1–10 have 4 hats total, and now we also know positions 1–8, 3 people wear hats.
So positions 9–10 contribute 1 hat
But we dont get any information about T from this.
We dont know how many people are possible in T, i.e we dont know how many people are there after position 10
Statement 2 is insufficient
A.Statement (1) alone is sufficient, but statement (2) alone is not.
4 of the first 10 people from the left are wearing hats
All the remaining people in the row are also wearing hats
Let the total number of people be T
Positions 1 through 10, 4 people wear hats
Positions 11 through T, all the people wear hats
Given statements,
(1) 5 of the 6 people from the right are wearing hats.
We know that, veryone from position 11 onward wears a hat, meaning only the first 10 may include non-hat wearers and now we also know that the last 6 people (from the right) include 5 wearing hats and 1 not wearing a hat
=> The last 6 people (positions T−5 to T) include some of the first 10 people (positions 1–10)
Now we want one of the last 6 people to be in the first 10
=> T- 5 <= 10
=> T <= 15
But we also want that, ONLY 1 of the last 6 people to be from position 1 - 10
=> T - 5 = 10
=> T = 15
So, lets try,
Positions 1–10 -> 4 hats, 6 no hats
Positions 11–15 -> all hats
For the last 6 people,
Position 10 = possibly not wearing a hat
Positions 11–15 = all hats which means 5 hats out of 6
So, T = 15 satisfies all the required conditions
Now,
If T < 15, more than one of the last 6 people would be in positions 1–10 which also contradicts the requirement
If T > 15, all of the last 6 people are beyond position 10 which contradicts the statement that only 5 of them wore hats.
So only possible value of T is 15.
(2) 3 of the 8 people from the left are wearing hats.
We know that, positions 1–10 have 4 hats total, and now we also know positions 1–8, 3 people wear hats.
So positions 9–10 contribute 1 hat
But we dont get any information about T from this.
We dont know how many people are possible in T, i.e we dont know how many people are there after position 10
Statement 2 is insufficient
A.Statement (1) alone is sufficient, but statement (2) alone is not.
08 Jul 2025, 07:54
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Given:
4 of the first 10 people from the left wear hats. (This means 6 of the first 10 do not wear hats).
All people after the 10th person wear hats. (This means any person not wearing a hat must be among the first 10).
Statement (1): 5 of the 6 people from the right wear hats.
This means 1 person among the last 6 does not wear a hat.
Since all non-hat wearers are among the first 10, this one non-hat wearer must be within the intersection of the first 10 people and the last 6 people.
If the row has N people, the last 6 people are P
N−5 ,...,PN
If N≤10, this statement contradicts the given. (e.g., if N=10, 4 hats in 10, but 5 hats in last 6, impossible). So N>10.
If N≥16, the last 6 people are all beyond P10 , so they all wear hats. This contradicts having 1 non-hat. So N<16.
Therefore, 10<N≤15. This gives possible values like 11, 12, 13, 14, 15.
Not sufficient.
Statement (2): 3 of the 8 people from the left wear hats.
This means 5 people among the first 8 do not wear hats (NH1−8 =5).
Combined with the given that 4 of the first 10 wear hats (NH1−10 =6):
The people not wearing hats among P9 and P10 must be 6−5=1.
So, exactly one of P9 or P10 does not wear a hat, and the other does.
This statement doesn't give information about the total number of people in the row (N).
Statement 2 is Not sufficient.
Combining (1) and (2):
From (1), N is between 11 and 15 (inclusive).
From (2), we know that one of P9 or P10 is the single non-hat wearer from that pair.
From (1), the single non-hat wearer among the last 6 people must be either P9 or P10 (because all people beyond P10 wear hats).
Let's test if we can find a unique N that satisfies both conditions for both possibilities (P9 is NH vs. P10 is NH).
Scenario A: P9 is the non-hat wearer (and P10 wears a hat).
If P9 is the non-hat wearer in the last 6, then P9 must be in positions PN−5 ,...,PN . This means N−5≤9⇒N≤14.
This scenario allows N∈{11,12,13,14}. All these values are consistent with the conditions.
Scenario B: P10 is the non-hat wearer (and P9 wears a hat).
If P10 is the non-hat wearer in the last 6, then P10 must be in positions P
N−5 ,...,PN . This means N−5≤10⇒N≤15.
This scenario allows N∈{11,12,13,14,15}. All these values are consistent with the conditions.
Since both scenarios are possible, and each scenario allows for multiple values of N (e.g., N=11 is possible in both), we cannot determine a unique value for N.
Conclusion: Even with both statements, the number of people in the row cannot be uniquely determined.
Answer : E
4 of the first 10 people from the left wear hats. (This means 6 of the first 10 do not wear hats).
All people after the 10th person wear hats. (This means any person not wearing a hat must be among the first 10).
Statement (1): 5 of the 6 people from the right wear hats.
This means 1 person among the last 6 does not wear a hat.
Since all non-hat wearers are among the first 10, this one non-hat wearer must be within the intersection of the first 10 people and the last 6 people.
If the row has N people, the last 6 people are P
N−5 ,...,PN
If N≤10, this statement contradicts the given. (e.g., if N=10, 4 hats in 10, but 5 hats in last 6, impossible). So N>10.
If N≥16, the last 6 people are all beyond P10 , so they all wear hats. This contradicts having 1 non-hat. So N<16.
Therefore, 10<N≤15. This gives possible values like 11, 12, 13, 14, 15.
Not sufficient.
Statement (2): 3 of the 8 people from the left wear hats.
This means 5 people among the first 8 do not wear hats (NH1−8 =5).
Combined with the given that 4 of the first 10 wear hats (NH1−10 =6):
The people not wearing hats among P9 and P10 must be 6−5=1.
So, exactly one of P9 or P10 does not wear a hat, and the other does.
This statement doesn't give information about the total number of people in the row (N).
Statement 2 is Not sufficient.
Combining (1) and (2):
From (1), N is between 11 and 15 (inclusive).
From (2), we know that one of P9 or P10 is the single non-hat wearer from that pair.
From (1), the single non-hat wearer among the last 6 people must be either P9 or P10 (because all people beyond P10 wear hats).
Let's test if we can find a unique N that satisfies both conditions for both possibilities (P9 is NH vs. P10 is NH).
Scenario A: P9 is the non-hat wearer (and P10 wears a hat).
If P9 is the non-hat wearer in the last 6, then P9 must be in positions PN−5 ,...,PN . This means N−5≤9⇒N≤14.
This scenario allows N∈{11,12,13,14}. All these values are consistent with the conditions.
Scenario B: P10 is the non-hat wearer (and P9 wears a hat).
If P10 is the non-hat wearer in the last 6, then P10 must be in positions P
N−5 ,...,PN . This means N−5≤10⇒N≤15.
This scenario allows N∈{11,12,13,14,15}. All these values are consistent with the conditions.
Since both scenarios are possible, and each scenario allows for multiple values of N (e.g., N=11 is possible in both), we cannot determine a unique value for N.
Conclusion: Even with both statements, the number of people in the row cannot be uniquely determined.
Answer : E
08 Jul 2025, 07:57
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A] If only 5 of the 6 are wearing hats then there must be 15 people. SUFFICIENT
B] this does not give us information about the people after the first 10. Insufficiency
Answer A.
B] this does not give us information about the people after the first 10. Insufficiency
Answer A.
Bunuel
