GMAT Club Olympics 2024 (Day 3): A basket contains apples and oranges

Question

A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another  without  replacement, what is the probability that at least one of them is an apple?

(1) The number of oranges in the basket is twice the number of apples.

(2) If 2 fruits are randomly selected from the basket one after another  with  replacement, the probability that at least one of them is an apple is 5/9.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket without replacement, what is the probability that at least one of them is an apple?

(1) The number of oranges in the basket is twice the number of apples.

This implies O = 2A. Different numbers of oranges and apples will give different probabilities. For example, if there are 2 oranges and 1 apple, then the probability of selecting at least one apple would be P(at least one apple) = 1 - P(no apples) = 1 - (2/3 * 1/2) = 2/3. However, if there are 4 oranges and 2 apples, then the probability of selecting at least one apple would be P(at least one apple) = 1 - P(no apples) = 1 - (4/6 * 3/5) = 3/5.

(2) If 2 fruits are randomly selected from the basket one after another  with  replacement, the probability that at least one of them is an apple is 5/9.

This implies 1 - O/(O + A) * O/(O + A) = 5/9:

(O/(O + A))^2 = 4/9

O/(O + A) = 2/3

3O = 2O + 2A

O = 2A.

As discussed above, such a ratio is not sufficient to get a unique answer. Not sufficient.

(1) + (2) Both statements give the same exact information: O = 2A. Thus, combining them does not add anything new. Not sufficient.

Answer: E.

8Harshitsharma · Answer

P(at least 1 apple) = 1 - P(no apple)
P(no apple) = \frac{	ext{no. of Oranges}}{	ext{no. Oranges + no. of Apples}}*\frac{	ext{no. of Oranges}}{	ext{no. Oranges + no. of Apples -1 }}

We need the value of no. of oranges and apples to answer the question,

Statement 1: Gives relation between oranges and apples but we don't know the exact value of apples and oranges to solve the equation.

Statment 2: It says P(at least 1 apple with replacement) = 5/9 which is equal to 
 \frac{	ext{no. of Oranges}}{	ext{no. Oranges + no. of Apples}}*\frac{	ext{no. of Oranges}}{	ext{no. Oranges + no. of Apples }}

Solving this we get relation between no. of oranges and no. of Apples but not the exact value. Again we get, Oranges = 2*Apples

Combining both statements:
We arrive at the same equation from both statements, so data is not sufficient to answer the question.

Answer is Choice E

Kinshook · Answer

A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another  without  replacement, what is the probability that at least one of them is an apple? 
(1) The number of oranges in the basket is twice the number of apples.
Let the number of oranges and number of apples be 2x & x respectively.
Total ways to select 2 fruits without replacement =  ^{3x}C_2 = \frac{3x(3x-1)}{2} 
Unfavorable ways (both fruits are oranges) =  ^{2x}C_2 = 2x(2x-1)/2 = x(2x-1) 
The probability that none of them is apple =  \frac{2x(2x-1)}{3x(3x-1)} 
The probability that at leaset one of them is apple =  1 - \frac{2x(2x-1)}{3x(3x-1)} 
Which depends on the value of x.
NOT SUFFICIENT

(2) If 2 fruits are randomly selected from the basket one after another  with  replacement, the probability that at least one of them is an apple is 5/9.Let the number of oranges and number of apples be o & a respectively.
Total ways to select 2 fruits with replacement = (o+a)^2
Unfavorable ways (both fruits are oranges) = o^2
The probability that none of them is apple = o^2/(o+a)^2
The probability that at leaset one of them is apple =1 - o^2/(o+a)^2 = 5/9
o^2/(o+a)^2 = 4/9
o/(o+a) = 2/3
Total ways to select 2 fruits without replacement = ^{o+a}C_2 = (o+a)(o+a-1)/2 
Unfavorable ways (both fruits are oranges) =  ^oC_2 = o(o-1)/2 
The probability that none of them is apple =  o(o-1)/(o+a)(o+a-1) 
The probability that at leaset one of them is apple =  1 - o(o-1)/(o+a)(o+a-1) = 1 - 2(o-1)/2(o+a-1) 
Which depends on values of o & a.
NOT SUFFICIENT

(1) + (2) 
(1) The number of oranges in the basket is twice the number of apples.
(2) If 2 fruits are randomly selected from the basket one after another  with  replacement, the probability that at least one of them is an apple is 5/9.
Let the number of oranges and number of apples be 2x & x respectively.
Total ways to select 2 fruits with replacement = (3x)^2
Unfavorable ways (both fruits are oranges) = (2x)^2
The probability that none of them is apple = (2x)^2/(3x)^2 = 4/9
The probability that at leaset one of them is apple =1 - 4/9 = 5/9

Total ways to select 2 fruits without replacement =  ^{3x}C_2 = \frac{3x(3x-1)}{2} 
Unfavorable ways (both fruits are oranges) =  ^{2x}C_2 = 2x(2x-1)/2 = x(2x-1) 
The probability that none of them is apple =  \frac{2x(2x-1)}{3x(3x-1)} 
The probability that at leaset one of them is apple =  1 - \frac{2x(2x-1)}{3x(3x-1)} 
Which depends on the value of x.
NOT SUFFICIENT

IMO E

Purnank · Answer

Given - A basket contains apples and oranges only.
Assume a for apples, and o for oranges.
The total number of fruits in the basket is = T = o + a

To find - If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple?

1st - The number of oranges in the basket is twice the number of apples.
o = 2a and T = 3a

If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple? 
Which is equal to = 1 - (Probabillity of no apples)
Probability of no apples = (o / 3a ) * ( o-1 / 3a-1) as we know o = 2a
Probability of no apples = (2a/3a) * (2a-1) / (3a-1) = (2*(2a-1)) / (3*(3a-1))
Therefore, Probability that at least one of them is an apple = 1 - (2*(2a-1)) / (3*(3a-1))
If we solve this we will get Probability that at least one of them is an apple = (5a-1) / (9a-3)
but we dont know a so we cant solve. Not sufficient.

2nd - If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is 5/9.
When fruits are selected with replacement, the probability of at least one apple in 2 selections is = 1 - (Probability of no apple) 
(Probability of no apple) = (o/3a)*(o/3a) 
Therefore 5/9 = 1 - (o/3a)^2
o/3a = 2/3
o/a = 2
o=2a which is same as per statement 1 hence from there as well we cant reach to conclusion. Not sufficient.

Even after combining we cant reach to conclusion.
Answer E.

paradise1234 · Answer

A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple?

P(at least one is apple) = 1- P(No apple) = 1 - P(Both Oranges) = ?

(1) The number of oranges in the basket is twice the number of apples. 
O= 2x
A=x
1 - P(Both Oranges) = 1- 2x/3x * (2x-1)/(3x-1)
 I is Insufficient.

(2) If 2 fruits are randomly selected from the basket one after another  with replacement , the probability that at least one of them is an apple is 5/9.

P(at least one is apple) = 1- P(No apple) = 1 - P(Both Oranges) = 5/9
1- Oranges/Total * Oranges/Total = 5/9
Oranges/Total * Oranges/Total = 4/9
Oranges/Total = 2/3
Therefore, Oranges = 2x, Apples=3x
1 - P(Both Oranges) = 1- 2x/3x * (2x-1)/(3x-1)
 II is Insufficient.

Combining both I and II will not give any result as both give the same information.
Ans E

pranjalshah · Answer

Answer: E

Statement (1):
The number of oranges in the basket is twice the number of apples.

Let $ A $ be the number of apples and $ O $ be the number of oranges.

From statement (1), we have:
\

The total number of fruits in the basket is:
\

The probability of selecting at least one apple can be found by subtracting the probability of selecting no apples (i.e., selecting two oranges) from 1.

The probability of selecting two oranges without replacement is:
\

Therefore, the probability of selecting at least one apple is:
\

Since $ A $ is an unknown integer, this expression alone does not give a specific numerical probability. Therefore, statement (1) alone is not sufficient to determine the probability.

Statement (2):

If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is $ \frac{5}{9} $.

With replacement, the probability of selecting at least one apple can be calculated as follows:

Let the total number of fruits be $ n $ and the number of apples be $ A $.

The probability of selecting an apple in one draw is:
\

The probability of not selecting an apple in one draw is:
\

The probability of not selecting an apple in both draws (with replacement) is:
\

Therefore, the probability of selecting at least one apple in two draws (with replacement) is:
\

Solving for $ \left( \frac{n-A}{n} \right)^2 = \frac{4}{9} $:
\ 
\ 
\ 
\ 
\

This implies the same relationship between the number of apples and oranges as given in statement (1): $ O = 2A $.

Since statement (2) also leads to the same conclusion about the ratio of apples to oranges, it does not independently provide sufficient additional information to determine the probability of selecting at least one apple without replacement.

Combining Statements (1) and (2):

Both statements confirm that the total number of fruits is $ 3A $ and the number of oranges is $ 2A $. However, combining these statements does not yield any new information beyond what each statement provides individually.

Thus, we still do not have enough information to determine the exact probability without the specific value of $ A $.

Therefore, the correct answer is:
E. Statements (1) and (2) TOGETHER are NOT sufficient.

0ExtraTerrestrial · Answer

To determine the probability we need to determine the number of apples and the total number of fruits in the basket, here since two successive selections are happening, without replacement, we'll need more information that just the initial ratio of total apples to total fruits. Let's look at the statements:

I. The number of oranges in the basket is twice the number of apples. 
Let number of oranges be x and number of apples be y, we're told that x=2y

Probability of getting atleast one apple in two picks = 1 - Probability of getting only oranges in two picks

Probability of getting the first orange= x/(x+y) = 2y/(y+2y)= 2y/3y = 2/3
Probability of getting the second orange= (x-1)/(x+y-1) = (2y-1)/(3y-1)
Thus probability of getting two oranges in 2 picks= 2(2y-1)/3(3y-1)
And probability of getting at least one apple in two picks= 1 - 2(2y-1)/3(3y-1)

Since we're not given either y or (2y-1)/3y-1), this statement is not sufficient to answer the question.

II. If 2 fruits are randomly selected from the basket one after another *with* replacement, the probability that at least one of them is an apple is 5/9.
Essentialy meaning that the probability of both of them being oranges is 1- 5/9 or 4/9.
This process is one with replacement, therefore, probability of an event happening twice= (probability of an event happening once)^2
Thus the probability of getting an orange on the first try= square root of (4/9) = 2/3

This means that the number of oranges and the total fruits are in the ratio 2:3. This is the exact same info that I was giving us, that wasn't sufficient so so isn't this, on the other hand II has also not provided us with any other information to complement I, so even together I and II are not sufficient to answer this question.

Posted from my mobile device

xhym · Answer

The probability that at least one is an apple is the same as  1 -  the probability that neither are apples.

In other words, it's  1 -  the probability that both are oranges.

1-P(both oranges)

1-\frac{o}{o+a}*\frac{o-1}{o+a-1}

1-\frac{o(o-1)}{(o+a)(o+a-1)}

Statement 1

We are told that  o=2a

1-\frac{2a(2a-1)}{(2a+a)(2a+a-1)}

1-\frac{4a^2-2a}{(3a)(3a-1)}

1-\frac{4a^2-2a}{9a^2-3a}

1-\frac{2a(2a-1)}{3a(3a-1)}

1-\frac{2(2a-1)}{3(3a-1)}

However, since we don't know a, we cannot find the probability.

-> Statement 1 alone is not enough.

Statement 2

If 2 fruits are randomly selected from the basket one after another  with  replacement, the probability that at least one of them is an apple is 5/9

Once again, the probability here is ( 1 -  the probability that both are oranges). However, there is replacement this time.

1-P(both orange) = \frac{5}{9}

P(both orange) = 1-\frac{5}{9}

P(both orange) = \frac{4}{9}

This means that  (\frac{o}{o+a})^2 = \frac{4}{9}

(\frac{o}{o+a}) = \frac{2}{3}

We now know the probabilty of selecting an orange in the first pick of the problem is  \frac{2}{3}  but we still don't know anything about the number of apples or oranges so we cannot find how many will be found without replacement.

-> Statement 2 alone is not enough.

Both statements together

Statement 1 told us that  o=2a  and from statement 2, we found that  (\frac{o}{o+a}) = \frac{2}{3}

Rewritten, statement 2 gives us:

3o = 2(o+a)

3o = 2o+2a

o = 2a

This is not new information so both statements together will not help either.

The answer is  E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

pianogirl · Answer

The value of the probibility of choosing at least one apple is equal to 1-the probability of choosing two oranges, so let's search that as it is simpler.

Let O be the number of oranges and A the number of apples. Let P(O^2) the probability of choosing two oranges without replacement and P'(O^2) the probability of choosing two oranges with replacement.

(1) The number of oranges in the basket is twice the number of apples.
if O = 2A

$P(O^2) = P(O1)*P(O2) = \frac{O}{O+\frac{1}{2}O} *\frac{O-1}{O-1+\frac{1}{2}O} = \frac{2}{3} * \frac{(2O-2)}{(3O-2)}$

without knowing the value of O it is not possible to deduce the value of $P(O^2)$. The statement (1) is insufficient

(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is 5/9.
let P' be the probability with replacement
$P'(O^2) = (\frac{O}{O+A})^2 = 1-\frac{5}{9} = \frac{4}{9} <=>\frac{O}{O+A} = \frac{2}{3} <=> 3O = 2O+2A <=> O = 2A $

The statement (2) is equivalent to statement (1) and is insufficient

Therefore the answer is E

Spectre26 · Answer

Let quantity of apples be  x  and oranges be  y 
Probability that 2 fruits are randomly selected, one after another without replacement:

Probability of getting  (orange first then apple + apple first then orange + getting apples twice): 
Probability:  \frac{x}{x+y}*\frac{y}{x+y-1} + \frac{y}{x+y}*\frac{x}{x+y-1} + \frac{x}{x+y}*\frac{x}{x+y-1}= \frac{(2xy+x^2)}{(x+y)(x+y-1)}

Let's analyse the statements:
(1) The number of oranges in the basket is twice the number of apples.
as per this,  y = 2x 
substituting y in the probability equation we get:
Probability =  \frac{(2x*2x+x^2)}{(x+2x)(x+2x-1)} = \frac{5x^2}{3x(3x-1)} 
We dont get any conclusive result here.

(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is 5/9.
If the fruits are selected with replacement, the probability is:
Probability of getting  (orange first then apple + apple first then orange + getting apples twice): 
Probability:  \frac{x}{x+y}*\frac{y}{x+y} + \frac{y}{x+y}*\frac{x}{x+y} + \frac{x}{x+y}*\frac{x}{x+y}= \frac{(2xy+x^2)}{(x+y)^2}

\frac{(2xy+x^2)}{(x+y)^2} = \frac{5}{9} 
We dont get any conclusive result here, only considering this statement.

If we use statement I with statement II, by subsituting no of oranges, we get:
 \frac{(2x*2x+x^2)}{(x+2x)^2} = \frac{5}{9} 
 \frac{5x^2}{9x^2} = \frac{5}{9} 
Here also, we dont get any conclusive result.

Correct Option: E

Lizaza · Answer

Regardless of taking the fruit with or without replacement, we will have two variables which we will need to find out:
 
 The probability of taking both apples =  \frac{a}{(a+o)} * \frac{a-1}{(a+o-1)}  
 The probability of taking the first apple =  \frac{a}{(a+o)} * \frac{o}{(a+o-1)}  
 The probability of taking the second apple =  \frac{o}{(a+o)} * \frac{a}{(a+o-1)}   
So as you see, even after tweaking these formulas, we will still need to know the values of these two vairables.
Even knowing the ratio of the two won't help us with the second fraction: for instance, with o=2a, we will need to calculate  \frac{a}{(a+2a-1)} = \frac{a}{3a-1} , and this is impossible without actually knowing the values.

Therefore, neither of the statements - because eventually both of them only give us the ratio of O to A - will be sufficient to find the final answer.
  The answer is E.

GuadalupeAntonia · Answer

A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple?
Apple =x
Oranges =y

P(at least one is apple)= 1- P(all oranges)

P(all oranges)= (y/(y+x))*((y-1)/(y+x-1)

P(at least one is apple)= 1-(y/(y+x))*((y-1)/(y+x-1) (EQ 1)

(1) The number of oranges in the basket is twice the number of apples.

y=2x
if we substitute in EQ 1, we have 2 incognitos 1 equation
NO SUFFICIENT

(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is 5/9.

P(at least one is apple)= 1- P(all oranges)

P(all oranges)= (y/(y+x))*((y)/(y+x)=5/9 (EQ2)
we have 2 incognitos 1 equation
NO SUFFICIENT

TOGETHER 1 AND 2

P(all oranges)= (y/(y+x))*((y)/(y+x)=5/9
from 1 we have that y=2x so if we substitute
in EQ2 from (2) we get a ratio so NOT SUFFICIENT

ANS LETTER  E

Suboopc · Answer

So essentially, we have to find the probability of selecting only oranges and subtract it from 1
which is 
1 -  \frac{O}{O+A}  *  \frac{O - 1}{O - 1 + A} 
O and A are number of oranges and number of apples respectively.

Statement 1: substituting O = 2A
1 -  \frac{2A}{3A}  *  \frac{2A - 1}{3A - 1}

We cannot solve this expression unless we know A. Insufficient.

Statement 2: 
1 -  (\frac{O}{O+A})^2  =  \frac{5}{9}

\frac{O}{O+A}  =  \frac{2}{3}

Substituting in expression that we have to calculate

1 -  \frac{2}{3}  *  \frac{O-1}{O-1+A}

We still cannot solve for the actual probability unless we know the values of O and A
Insufficient.

Both together:
we end up with the equation in the first case
1 -  \frac{2}{3}  *  \frac{2A - 1}{3A - 1}

Insufficient.

E

smile2 · Answer

We are asked to find the probability of selecting none 🍎 (apples) and subtract it from 1
which is 
1 -  \frac{G}{G+A}  *  \frac{G - 1}{G - 1 + A} 
G and A are number of 🍊 (oranges) and number of 🍎 (apples) respectively.

Statement 1: substituting G = 2A
1 -  \frac{2A}{3A}  *  \frac{2A - 1}{3A - 1}

A alone is Insufficient.

Statement 2:  
1 -  (\frac{G}{G+A})^2  =  \frac{5}{9} 
 \frac{G}{G+A}  =  \frac{2}{3}

Substituting in expression that we have to calculate

1 -  \frac{2}{3}  *  \frac{G-1}{G-1+A}

B alone is Insufficient as we don't know the values of G and A

Both together
1 -  \frac{2}{3}  *  \frac{2A - 1}{3A - 1} 
Both combined is insufficient as well.

Therefore the right answer choice is E

Catman · Answer

(1) The number of oranges in the basket is twice the number of apples. 
Apples = 2, Oranges =4
Apples = 4, Oranges =8

From this, we get different values for the probability.

(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is 5/9. 
Probability that at least one of them is an apple = 1- Probability of Both oranges without replacement
Probability of both oranges = 1- \frac{5}{9} = \frac{4}{9}  
 \frac{2}{3} *\frac{2}{3} 
This means out of total 3x fruits 2x are oranges and x are apples.
2 oranges 1 apple and 3 total.
the probability that at least one of them is an apple = 1- Probability of Both oranges without replacement

Probability of Both oranges without replacement
=  \frac{2}{3}*\frac{1}{2} = 0.33 
4 Oranges 2 apples and 6 total.
Probability of Both oranges without replacement =  \frac{4}{6}*\frac{3}{5}=0.4 
Using this also we cannot determine the probabiltiy.

Combining the 2 statements gives no additional information.

IMO E.

d_patel · Answer

Question stem provides us with following information:

A basket contains apples and oranges.
And we are selecting 2 fruits randomly from basket one after another without replacement.

We need to find probability that at least one of them is an apple.

Statment-1 
The number of oranges in the basket is twice the number of apples.
So if number of apples are  x , then number of oranges are  2x .
Total fruits in basket  = x + 2x = 3x

There is only 1 case when at least one of the fruit will be not apple -> if both fruits are orange. 
So, we can find probability of selecting both fruits of orange and then substract it from 1.

Probability that 1st fruit is orange =  \frac{2x}{3x}  =  \frac{2}{3} 
Probability that 2nd fruit is orange given first fruit was orange =  \frac{2x-1}{3x-1}

So the probability that both fruits are orange =  \frac{2}{3}  *  \frac{2x-1}{3x-1}

Then, the probability that at least one fruit is an apple is:
1 -  \frac{2}{3}  *  \frac{2x-1}{3x-1}

We cannot find probability based on this equation as value of x is unknown.

Statement-1 is not sufficient.

Statment-2 
If  p  is probability of selecting an apple then  1-p  will be the probability of selecting an orange as we are replacing the fruit.

Then as we established in statement-1, the probability of selecting atleast 1 apple can be found by first finding probability of selecting 2 oranges and then subtracting it from 1.
 1 - (1-p)^2  =  \frac{5}{9} 
 (1-p)^2 = 4/9 
 1-p = 2/3 
 p = 1/3 
 
So we can say that 1/3 of the fruits are apples and 2/3 are oranges.

If total number of apples are x, then oranges will  be 2x.
Total fruits will be 3x.
We reached same point as statement-1.
We will get same equation.

So, this statement is also not sufficient.

Combining statement-1 and statement-2: 
We reached same equations from both statement. 
So, both statements are also not suffice.

Final answer - E

Gp111 · Answer

Statement 1 
From this statement alone, we know the ratio of oranges to apples, but we do not know the total number of fruits. Without the total number of fruits, we cannot calculate the exact probability of selecting at least one apple when two fruits are selected without replacement. Therefore,  Statement 1 alone is not sufficient

Statement 2 
This statement gives us the probability of selecting at least one apple in two draws with replacement. From this, we can infer the ratio of apples to total fruits. This confirms that the ratio ( O = 2A ).
 1 - \frac{O}{A + O}^2 = \frac{5}{9} 
However, like Statement 1, it does not give us the exact number of apples or oranges, only the ratio. Therefore,  Statement 2 alone is not sufficient

Combining Both Statements 
Both statements confirm the ratio ( O = 2A ). However, without knowing the exact number of apples or oranges, we cannot determine the exact probability of selecting at least one apple when two fruits are drawn without replacement

Both statements together confirm the ratio of apples to oranges but do not provide the exact numbers needed to calculate the probability without replacement. Therefore, the correct answer is:  E

Mohit11111 · Answer

Someone guide where I am going wrong,

Possibilities of selecting at least one apple = 3cases
1. Orange, Apple
2. Apple, Orange
3. Apple, Apple

As per Statement 1, O=2A let say apples is x, oranges =2x , Total = 3x

In statement 2 we add replacement, now,
Case 1 = (2x * x)/3x * 3x

case 2= (x * 2x)/3x * 3x
Case 3= (x * x)/3x * 3x
Adding all three cases we get 5/9

Why is this wrong

Bunuel · Answer

You stopped halfway. Case 1 gives 2/9, case 2 gives 2/9, and case 3 gives 1/9. Adding these gives 5/9. So? You cannot find x.

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