GATE CSE 2017 | Set 2 | GA | Question: 7

Question

There are three boxes. One contains apples, another contains oranges and the last one contains both apples and oranges. All three are known to be incorrectly labeled. If you are permitted to open just one box and then pull out and inspect only one fruit, which box would you open to determine the contents of all three boxes?

• A. The box labeled ‘Apples’
• B. The box labeled ‘Apples and Oranges’
• C. The box labeled ‘Oranges’
• D. Cannot be determined

Comments

Since we want none of the boxes with its correct label. Therefore, the boxes can be labelled in $D_3$ ways. Where $D_3=$ Derangement of $3$ elements which is equal to $2$. Write down both configuration of derangemennt and then check option $A,B,C$.

Another way of thinking is that since its an mcq question only $1$ option can be correct. Option $A$ and $C$ can be ruled out because if any one of them is the answer, then the other must be the answer as well. For option $B$ check apply it to the above configuration and see if it is possible to get the answer or not.

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NOTE: Please everyone check this solution and tell if I got this wrong(or maybe am I not understanding the solution) or the question is debatable or a bit flawed in it as any boxes can be used?

🧠 3-box trick — nutshell

        ALL 3 LABELS WRONG ❌
                 │
      ┌──────────┼──────────┐
      ↓          ↓          ↓
   🍎 BOX     🍊 BOX     🍎+🍊 BOX
      │          │          │
      ↓          ↓          ↓
   Pick 1     Pick 1      Pick 1
    fruit       fruit       fruit
      │          │          │
      └──────────┼──────────┘
                 ↓
       🔍 Identify that box
                 ↓
      🔄 Remaining 2 are forced
                 ↓
              ✅ DONE

⚡ All 3 work

Box opened	What happens
📦 “Apples”	🍎/🍊 drawn → box identified → remaining 2 forced
📦 “Oranges”	🍎/🍊 drawn → box identified → remaining 2 forced
📦 “Apples + Oranges”	Must be pure ❌🍎+🍊 → 1 fruit identifies it → remaining 2 forced

🎯 Conclusion

ANY ONE of the 3 boxes can be opened​



In other words please do note that:

📦 We have 3 actual possibilities

• 🍎 = Apples only
• 🍊 = Oranges only
• 🍎🍊 = Mixed

And every label is wrong ❌.

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1️⃣ Open 📦 “Apples”

Suppose you pick 🍎.

Since the label is wrong, this box cannot be Apples only.

So it must be:

📦 “Apples”=🍎🍊

Now only 🍎 and 🍊 remain for the other two boxes.

• 📦 “Oranges” cannot be Oranges → must be 🍎
• 📦 “Apples + Oranges” → must be 🍊

✅ Everything determined.

If you picked 🍊 instead, the same logic works—the possibilities simply swap.

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2️⃣ Open 📦 “Oranges”

Exactly symmetrical.

Suppose you pick 🍊:

📦 “Oranges”=🍎🍊

Then:

📦 "Apples"          → 🍊
📦 "Apples + Oranges" → 🍎

✅ Everything determined.

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3️⃣ Open 📦 “Apples + Oranges”

This one is even easier.

Its label is wrong, so:

📦 “Apples + Oranges”≠🍎🍊

Therefore it must be pure:

🍎OR🍊

Pick one fruit.

🍎 picked → box = 🍎 only
🍊 picked → box = 🍊 only

Then the remaining two boxes are automatically determined.

✅ Everything determined.

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🧠 Final visualization

ALL LABELS WRONG ❌
        │
        ├── 📦 Apples ──→ pick 1 fruit → 🔍 identify → remaining 2 forced
        │
        ├── 📦 Oranges ─→ pick 1 fruit → 🔍 identify → remaining 2 forced
        │
        └── 📦 Mixed ───→ cannot be mixed → 🍎/🍊 → 🔍 identify → remaining 2 forced

🎯 Hence

All three boxes work. Since the question asks “which box” uniquely, no single option can be selected.

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See to this if it is right or wrong please inform

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@AJ_@G27 

The Failure: Opening the box labeled "Apples"

Because the label is lying, this box cannot contain just apples. It has only two options for what is actually inside:

• Option 1: It is full of only Oranges (🍊).
• Option 2: It is a Mixed box (🍎 + 🍊).

Now, you reach in with your eyes closed and pull out an Orange (🍊).

The Trap: Which of the two options is it?

• Is it the box with only Oranges? Yes, that would give you an orange.
• Is it the Mixed box? Yes, you could have just happened to grab an orange out of the mix.

You are completely stuck. You cannot decide between Option 1 or Option 2.

Answer 1

B. the box labelled 'Apples and Oranges'.

Reason.

We know that the box labeled "Apples and Oranges" can't contain both, so whatever we pick will be the correct label.

Without the loss of generality say picked item was an orange, what that means is that the box that is labeled "Apples" can't contain just oranges. It also can't contain just apples as it is known to be wrong. So it only can contain combination of oranges and apples. Now the third box labeld "Oranges" contains only apples.

Comments

thanks , I missed the line " inspect only one fruit"

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@Dhruv Patel very clear and concise answer.

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Great explanation!

Answer 2

Open the box that is labeled "Apples and Oranges".You know that since none of the labels are correct, the box must either contain only apples, or only orangesSuppose, that you remove an apple from that box. Therefore, that box is the "Apples Only" box.One of the two remaining boxes must be the "Oranges Only" box. However, one of them is labeled "Apples Only", and the other is labeled "Oranges Only". Therefore, the one labeled "Apples Only" is the box that contains only oranges, and the box labeled "Oranges Only" is the box that has both kinds of fruit.

So ans is b

Answer 3

option B --both

lets assume:
box 1 is labelled Oranges (O)
box 2 is labelled Apples (A)
box 3 is labelled Apples and Oranges (A+O)
and that ALL THREE BOXES ARE LABELLED INCORRECTLY"

Pick a fruit from box 1,

1) if you pick an Orange:
- box 1's real label can only be O or A+O
- box 1's current label is O
- since ALL LABELS ARE INCORRECT then box 1's real label can not be O
- box 1's new label should then be A+O by elimination
- since ALL LABELS ARE INCORRECT
- box 2's label is changed to O
- box 3's label is changed to A
- SOLVED

2) if you pick an Apple:
- box 1's real label can only be A or A+O
- box 1's current label is O
- since ALL LABELS ARE INCORRECT then box 1's real label can not be O
- this still leaves us with the choice between label A and label A+O - which would both be correct
- FAILURE

Solution: The trick is to actually pick a fruit from the A+O labeled box

Pick a fruit from box 3:

1) if you pick an Orange:
- box 3's real label can only be O or A
- box 3's current label is A+O
- since ALL LABELS ARE INCORRECT then box 3's real label can not be A+O
- box 3's new label should then be O by elimination
- since ALL LABELS ARE INCORRECT
- box 1's label is changed to A
- box 2's label is changed to A+O
- SOLVED

2) if you pick an Apple:
- box 3's real label can only be O or A
- box 3's current label is A+O
- since ALL LABELS ARE INCORRECT then box 3's real label can not be A+O
- box 3's new label should then be A by elimination (not O)
- since ALL LABELS ARE INCORRECT
- box 1's label is changed to A+O
- box 2's label is changed to O
- SOLVED

link-https://www.glassdoor.co.in/Interview/There-are-three-boxes-one-contains-only-apples-one-contains-only-oranges-and-one-contains-both-apples-and-oranges-The-b-QTN_114922.html

Comments

@swap_it well-explained and detailed solution.

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since ALL LABELS ARE INCORRECT
- box 1's label is changed to A - How to know whether this is going to be O or A+O?
- box 2's label is changed to A+O - This could've been A, if orange was picked from 3rd box,

i'm not able to understand

Answer 4

Option should be B. The question has came aready in apple. Inc interview.

It's easier to draw it out.

There are only 2 possible combinations when all labels are tagged incorrectly.

All you need to do is pick one fruit from the one marked "Apples + Oranges".

If it's Apple, then change "Apple + Orange" to "Apple"
The "Apple" one change to "Orange"
The "Orange one change to "Apple + Orange"

If it's Orange, then change "Apple + Orange" to "Orange"
The "Apple" one change to "Apple + Orange"
The "Orange" one change to ""Apple"

source: https://www.glassdoor.co.in/Interview/There-are-three-boxes-one-contains-only-apples-one-contains-only-oranges-and-one-contains-both-apples-and-oranges-The-b-QTN_114922.htm

Comments

thx

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@vishwa ratna clear and easily understood solution.

Answer 5

B> is correct.

Assuming single option.

->Apple or orange cannot be selected .Since both can  be possible.

->Now only choice to assume your trick on option B.

->You already know how to proceed further.This question is already answered by DHRUV patel

Answer 6

1. The box labeled ‘Apples and Oranges   . if we take out any one of them then we get know that other one will be orange or apple