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 taps did Clive receive for his question?

The correct answer is 7.

Let's break down the rules of the game step-by-step to find the number of taps Clive received.

First, we need to determine the possible number of taps any single question can receive. There are 5 people playing the game, and each person asks a question to the other 4 people. Therefore, each question receives 4 responses.

The possible responses and their corresponding tap counts are:

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

We are told that each question received at least one "Yes", one "No", and one "Maybe". This accounts for 3 of the 4 responses. The sum of these three responses is:

$1+2+3=6$

The 4th response can be any of the three options (1, 2, or 3 taps). Adding this to the sum of the first three responses, the total number of taps received for any question can only be one of the following:

- If the 4th answer is "Yes": $6+1=7$ taps
- If the 4th answer is "No": $6+2=8$ taps
- If the 4th answer is "Maybe": $6+3=9$ taps

So, every question received exactly 7, 8, or 9 taps.

Now, let's look at the given information to find the distribution of the 40 total taps across the 5 questions.

From statement 1, we know Alia received exactly 9 taps to her question.

From statement 3, we know that Badal, Dilshan, and Ehsaan all received an equal number of taps to their respective questions. Let's represent this unknown number with the variable $y$.

Let's use the variable $x$ to represent the number of taps Clive received for his question.

Since the total number of taps heard across all 5 questions is 40, we can write the following equation for the sum of taps received by all five individuals (Alia + Clive + Badal + Dilshan + Ehsaan = 40):

$9+x+y+y+y=40$

Simplifying the equation, we get:

$x+3y=31$

We already established that the number of taps received by any question (including $x$ and $y$) can only be 7, 8, or 9. Let's test the possible values for $y$ to see which one gives a valid value for $x$:

- If $y=7$: $x+3\left(7\right)=31$, which means $x+21=31$. This gives $x=10$. This is impossible because the maximum number of taps a question can receive is 9.

- If $y=8$: $x+3\left(8\right)=31$, which means $x+24=31$. This gives $x=7$. This is a perfectly valid number of taps.

- If $y=9$: $x+3\left(9\right)=31$, which means $x+27=31$. This gives $x=4$. This is impossible because the minimum number of taps a question can receive is 7.

Therefore, the only valid scenario is when $y=8$ and $x=7$. Since $x$ represents the number of taps Clive received, we can conclude that Clive received exactly 7 taps to his question.