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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
28% (03:10) correct
72%
(02:55)
wrong
based on 25
sessions
History
Date
Time
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Not Attempted Yet
Adrián and Carlota are siblings, and they both have one banknote of each of 6 denominations: €10, €20, €50, €100, €200, and €500. If Carlota takes one of Adrián’s notes at random and subsequently gives Adrián one of the notes she then has at random, what is the probability that neither sibling will have more than €100 more than the other?
(A) \(\frac{1}{3}\)
(B) \(\frac{8}{21}\)
(C) \(\frac{3}{7}\)
(D) \(\frac{10}{21}\)
(E) \(\frac{2}{3}\)
(A) \(\frac{1}{3}\)
(B) \(\frac{8}{21}\)
(C) \(\frac{3}{7}\)
(D) \(\frac{10}{21}\)
(E) \(\frac{2}{3}\)
11 Aug 2026, 21:28
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Initially, both siblings have €880, so only the net transfer matters.
Suppose Carlota takes Adrian's note of value a, then gives Adrian a note of value g.
Final amounts:
Adrian: 880−a+g
Carlota: 880+a−g
Difference:
∣(880+a−g)−(880−a+g)∣=2∣a−g∣
=> 2∣a−g∣≤100
=> ∣a−g∣≤50
Count favorable outcomes: 6×7=42 equally likely outcomes. (The second step has 7 notes because Carlota now has her original 6 notes plus Adrian's note.)
Total favorable outcomes: 4+4+5+3+2+2=20
Ans = 20/42 = 10/21 (D)
Suppose Carlota takes Adrian's note of value a, then gives Adrian a note of value g.
Final amounts:
Adrian: 880−a+g
Carlota: 880+a−g
Difference:
∣(880+a−g)−(880−a+g)∣=2∣a−g∣
=> 2∣a−g∣≤100
=> ∣a−g∣≤50
Count favorable outcomes: 6×7=42 equally likely outcomes. (The second step has 7 notes because Carlota now has her original 6 notes plus Adrian's note.)
| Adrian's note "a" | Favorable notes Carlota can give |
| 10 | 10 (2 notes), 20, 50 → 4 |
| 20 | 10, 20 (2 notes), 50 → 4 |
| 50 | 10, 20, 50 (2 notes), 100 → 5 |
| 100 | 50, 100 (2 notes) → 3 |
| 200 | 200 (2 notes) → 2 |
| 500 | 500 (2 notes) → 2 |
Total favorable outcomes: 4+4+5+3+2+2=20
Ans = 20/42 = 10/21 (D)
12 Aug 2026, 01:27
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Taking an example is a good way to get some insight into a problem. What if the notes that they pick at random are the same? The amount of money that each has won't change, i.e. neither sibling will have more than 100 euros more than the other. However, if one picks a 200 euro note and the other a 20 euro note, then one sibling will have 180 more than the other.
For probability problems, we want to find the (desired outcomes)/(total outcomes), so it's worth thinking about what the total number of situations is. Since each person has 6 notes, then we might think that there are 36 total outcomes. But there's a twist here: the problem states that Carlota takes a note and then gives back 'one of the notes she then has', i.e. out of the 7 notes in her hand, two of which will be the same. This means that we have 6x7 = 42 possible outcomes.
At this point a good thought would be: do I need to consider each outcome separately? Perhaps we do, but that would take a long time. Looking for a pattern, or finding a way to estimate is usually helpful in probability problems. Given that most of the notes are less than 100 euros, we'd expect that most of the transfers end up in difference of less than 100 euros between the siblings, and that we'd be looking for a higher answer.
To solve this more precisely, we need to consider all the outcomes.
If Carlota takes a 10 euro note, then she can give Adrian back the following notes: 10, 10, 20, or 50 (i.e. 4 out of 7) and the difference will be less than 100. It probably helps to take another example here: if Carlota takes a 10 euro note then gives away a 50 euro note, then she will be 40 euros poorer, and Adrian will be 40 euros richer, giving a difference of 80 euros between them.
If Carlota takes a 20 euro note, then she can give Adrian back the following notes: 10, 20, 20, 50 (i.e. 4 out of 7) and the difference will be less than 100.
If Carlota takes a 50 euro note, then she can give Adrian back the following notes: 10, 20, 50, 50, 100 (i.e. 5 out of 7) and the difference will be less than 100. Another trap here: it's okay for one sibling to be 50 euros poorer and the other 50 euros richer - this gives a difference of 100 euros, but it still falls under the definition of 'neither sibling has more than 100 euros more than the other'.
If Carlota takes a 100 euro note, then she can give Adrian back the following notes: 50, 100, 100 (i.e. 3 out of 7) and the difference will be less than 100.
If Carlota takes a 200 euro note, then she can give Adrian back the following notes: 200, 200 (i.e. 2 out of 7) and the difference will be less than 100.
If Carlota takes a 500 euro note, then she can give Adrian back the following notes: 500, 500 (i.e. 2 out of 7) and the difference will be less than 100.
Adding up these outcomes, we have 20/42 = 10/21 as the answer. In conclusion, this is a very hard probability problem which is difficult to conceptualise and long to solve, with several traps along the way. Unless you're aiming for a very high score (85+), I'd recommend guessing on this problem and using your time elsewhere.
For probability problems, we want to find the (desired outcomes)/(total outcomes), so it's worth thinking about what the total number of situations is. Since each person has 6 notes, then we might think that there are 36 total outcomes. But there's a twist here: the problem states that Carlota takes a note and then gives back 'one of the notes she then has', i.e. out of the 7 notes in her hand, two of which will be the same. This means that we have 6x7 = 42 possible outcomes.
At this point a good thought would be: do I need to consider each outcome separately? Perhaps we do, but that would take a long time. Looking for a pattern, or finding a way to estimate is usually helpful in probability problems. Given that most of the notes are less than 100 euros, we'd expect that most of the transfers end up in difference of less than 100 euros between the siblings, and that we'd be looking for a higher answer.
To solve this more precisely, we need to consider all the outcomes.
If Carlota takes a 10 euro note, then she can give Adrian back the following notes: 10, 10, 20, or 50 (i.e. 4 out of 7) and the difference will be less than 100. It probably helps to take another example here: if Carlota takes a 10 euro note then gives away a 50 euro note, then she will be 40 euros poorer, and Adrian will be 40 euros richer, giving a difference of 80 euros between them.
If Carlota takes a 20 euro note, then she can give Adrian back the following notes: 10, 20, 20, 50 (i.e. 4 out of 7) and the difference will be less than 100.
If Carlota takes a 50 euro note, then she can give Adrian back the following notes: 10, 20, 50, 50, 100 (i.e. 5 out of 7) and the difference will be less than 100. Another trap here: it's okay for one sibling to be 50 euros poorer and the other 50 euros richer - this gives a difference of 100 euros, but it still falls under the definition of 'neither sibling has more than 100 euros more than the other'.
If Carlota takes a 100 euro note, then she can give Adrian back the following notes: 50, 100, 100 (i.e. 3 out of 7) and the difference will be less than 100.
If Carlota takes a 200 euro note, then she can give Adrian back the following notes: 200, 200 (i.e. 2 out of 7) and the difference will be less than 100.
If Carlota takes a 500 euro note, then she can give Adrian back the following notes: 500, 500 (i.e. 2 out of 7) and the difference will be less than 100.
Adding up these outcomes, we have 20/42 = 10/21 as the answer. In conclusion, this is a very hard probability problem which is difficult to conceptualise and long to solve, with several traps along the way. Unless you're aiming for a very high score (85+), I'd recommend guessing on this problem and using your time elsewhere.
