DIRECTIONS for questions: Read the information given below and answer the question that follows.

Alia, Badal, Clive, Dilshan, and Ehsaan played a game in which each asks a unique question to all the others and they respond by tapping their feet, either once or twice or thrice. One tap means “Yes”, two taps mean “No”, and three taps mean “Maybe”.

A total of 40 taps were heard across the five questions. Each question received at least one “Yes”, one “No”, and one “Maybe.”

The following information is known.

1. Alia tapped a total of 6 times and received 9 taps to her question. She responded “Yes” to the questions asked by both Clive and Dilshan.

2. Dilshan and Ehsaan tapped a total of 11 and 9 times respectively. Dilshan responded “No” to Badal.

3. Badal, Dilshan, and Ehsaan received equal number of taps to their respective questions.

4. No one responded “Yes” more than twice.

5. No one’s answer to Alia’s question matched the answer that Alia gave to that person’s question. This was also true for Ehsaan.

6. Clive tapped more times in total than Badal.

How many “Yes” responses were received across all the questions?

The correct answer is 6.

To solve this problem, we need to analyze the distribution of taps (responses) across the questions asked by the five individuals. Let's break down the information step-by-step.

First, we define the values of each response:

"Yes" = 1 tap
"No" = 2 taps
"Maybe" = 3 taps

Each of the 5 people asks 1 unique question to the other 4 people. This means every question receives exactly 4 responses.

The problem states that each question received at least one "Yes", one "No", and one "Maybe". Let's calculate the minimum number of taps for these three mandatory responses:

$1+2+3=6$ taps

Since each question has 4 responses, the 4th response must be either a "Yes" (1 tap), a "No" (2 taps), or a "Maybe" (3 taps). Therefore, the possible total number of taps a single question can receive are:

1. If the 4th response is "Yes": $6+1=7$ taps (contains 2 "Yes" responses)
2. If the 4th response is "No": $6+2=8$ taps (contains 1 "Yes" response)
3. If the 4th response is "Maybe": $6+3=9$ taps (contains 1 "Yes" response)

Now, let's look at the specific information provided for each person. Statement 1 tells us that Alia received 9 taps to her question. Statement 3 tells us that Badal, Dilshan, and Ehsaan received an equal number of taps to their respective questions. Let's call this unknown number of taps $x$.

We know the total number of taps across all 5 questions is 40. We can set up an equation:

$\mathrm{Taps(Alia)}+\mathrm{Taps(Badal)}+\mathrm{Taps(Clive)}+\mathrm{Taps(Dilshan)}+\mathrm{Taps(Ehsaan)}=40$

$9+x+\mathrm{Taps(Clive)}+x+x=40$

$3x+\mathrm{Taps(Clive)}=31$

Since the total taps for any question can only be 7, 8, or 9, we can test the possible values for $x$ (the number of taps received by Badal, Dilshan, and Ehsaan):

If $x=7$: $3\left(7\right)+\mathrm{Taps(Clive)}=31$ → $\mathrm{Taps(Clive)}=10$. This is not possible because the maximum taps a question can receive is 9.

If $x=8$: $3\left(8\right)+\mathrm{Taps(Clive)}=31$ → $\mathrm{Taps(Clive)}=7$. This is perfectly valid since 7 is a possible total.

If $x=9$: $3\left(9\right)+\mathrm{Taps(Clive)}=31$ → $\mathrm{Taps(Clive)}=4$. This is not possible because the minimum taps a question can receive is 7.

Thus, we have determined the total taps received by each person's question:

- Alia: 9 taps (contains 1 "Yes" response)
- Badal ($x$): 8 taps (contains 1 "Yes" response)
- Clive: 7 taps (contains 2 "Yes" responses)
- Dilshan ($x$): 8 taps (contains 1 "Yes" response)
- Ehsaan ($x$): 8 taps (contains 1 "Yes" response)

Finally, we sum the number of "Yes" responses received across all questions:

$1+1+2+1+1=6$

Therefore, a total of 6 "Yes" responses were received across all the questions.