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805+ (Hard)|   Combinations|      
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kevincan
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To earn her degree, Eugenia had to answer a sequence of at m 16 Mar 2026, 16:13
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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:

33% (02:56) correct 67% (02:25) wrong based on 39 sessions

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To earn her degree, Eugenia had to answer a sequence of at most ten questions. She was asked the first question. After that, she was asked another question only if at least half of her previous answers were correct.
If Eugenia was asked exactly seven questions, how many possible sequences of correct and incorrect answers are there?

A. 5
B. 6
C. 7
D. 8
E. 10
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A
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To earn her degree, Eugenia had to answer a sequence of at m 17 Mar 2026, 02:23
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If she were asked exactly 7 questions what must be true about her responses?
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Dereno
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To earn her degree, Eugenia had to answer a sequence of at m 17 Mar 2026, 12:38
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kevincan
To earn her degree, Eugenia had to answer a sequence of at most ten questions. She was asked the first question. After that, she was asked another question only if at least half of her previous answers were correct.
If Eugenia was asked exactly seven questions, how many possible sequences of correct and incorrect answers are there?

A. 5
B. 6
C. 7
D. 8
E. 10
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The question is stopped at q7, which means for q7 to occur, the previous 6 responses should be 3 correct and 3 wrong.

Because, if we have 4 Correct, then the possibility to ask 8th question remains valid.

For the second question to occur, the first question has to be correct only.

So, for question 5 to be asked. We need to have at least 2 or 3 correct.

I am mentioning the outputs in sequence (q1, q2,q3, q4)

For first two questions, the outputs are:

c c

c w

For the third question, the output can be:

c c c

c c w

c w c

c w w

For the fourth question, we need more than 1.5 questions to be correct. So , c w w is out.

c c c c ( this violates the case of max 3 c only).

c c c w

c c w c

c c w w

c w c c

c w c w

For Q 6 to occur, we need 2.5 outputs to be correct. So, c w c w is ruled out. So we have

c c c w w

c c w c w

c c w w (c/w)

c w c c w.


The possible cases after q5 are:

c c c w w

c c w c w

c c w w c

c c w w w

c w c c w

For q 7 to occur, we need 3 correct outcomes.

c c c w w w

c c w c w w

c c w w c w

c c w w w c

c w c c w w

The seventh outcome is definitely wrong. As this ends the continuation.

Hence, we have 5 outcomes.

option A

There must be an easier way to arrive at the answer. !! If I come across will share it.
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To earn her degree, Eugenia had to answer a sequence of at m 17 Mar 2026, 18:38
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Let me walk through this step by step.

Eugenia gets the next question only if at least half her previous answers are correct. She was asked exactly 7 questions, meaning she qualified for questions 2 through 7 but did NOT qualify for question 8.

Let's track how many correct answers she needs at each stage:

- After Q1: need at least 1/1 correct → Q1 must be CORRECT
- After Q2: need at least 1/2 correct (already guaranteed)
- After Q3: need at least 2/3 correct
- After Q4: need at least 2/4 correct
- After Q5: need at least 3/5 correct
- After Q6: need at least 3/6 correct
- After Q7: must have FEWER than 4/7 correct (so she stops)

Working backwards from Q7:

Since she needs at least 3 correct after Q6, but at most 3 correct after Q7, we get: exactly 3 correct after Q6, and Q7 is WRONG.

Since she needs at least 3 correct after Q5, and exactly 3 after Q6, that means: exactly 3 correct after Q5, and Q6 is WRONG.

So we know: Q1 = CORRECT, Q6 = WRONG, Q7 = WRONG, and exactly 3 total correct in Q1-Q5.

Since Q1 is correct, exactly 2 of Q2-Q5 must be correct.

Key Insight: The only remaining constraint is: after Q3, at least 2 must be correct. Since Q1 is correct, this means at least 1 of Q2 or Q3 must be correct.

Total ways to pick 2 correct out of Q2-Q5: C(4,2) = 6
Minus the 1 bad case where Q2 and Q3 are both wrong (only Q4, Q5 correct): 6 - 1 = 5

The [b]5 valid sequences for (Q2,Q3,Q4,Q5) are:[/b]
C,C,I,I | C,I,C,I | C,I,I,C | I,C,C,I | I,C,I,C

Answer: A (5)
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