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.

Which two people tapped an equal number of times in total?

The correct answer is Option 4.

We can determine the answer step-by-step by calculating the total number of taps made by each person in the game.

First, we are given that the total number of taps heard across all five questions is 40.

From the provided statements, we can extract the exact number of times three of the individuals tapped:

• Alia tapped 6 times (from statement 1).
• Dilshan tapped 11 times (from statement 2).
• Ehsaan tapped 9 times (from statement 2).

We can find the combined taps of the remaining two people, Badal and Clive, by subtracting the known taps from the total 40 taps:

$\mathrm{Badal}+\mathrm{Clive}=40-(6+11+9)$

$\mathrm{Badal}+\mathrm{Clive}=40-26=14$

Next, we need to figure out the minimum number of times any single person can tap. Each person is asked a question by the other 4 people, so they must answer 4 times. The available responses are 1 tap ("Yes"), 2 taps ("No"), and 3 taps ("Maybe").

Statement 4 gives us a crucial constraint: "No one responded 'Yes' more than twice." This means a person can give at most two 1-tap answers. To find the absolute minimum number of taps a person could possibly make across their 4 answers, they would need to use their two 1-tap answers, and then use the next lowest number (2 taps) for the remaining two answers.

Therefore, the minimum number of taps for any person is:

$1+1+2+2=6\text{taps}$

Since Badal and Clive together tapped 14 times, and we now know neither could have tapped fewer than 6 times, there are only three mathematical possibilities for their taps:

• Badal = 6 taps, Clive = 8 taps
• Badal = 7 taps, Clive = 7 taps
• Badal = 8 taps, Clive = 6 taps

Statement 6 specifies that "Clive tapped more times in total than Badal." This eliminates the combinations where their taps are equal (7 and 7) or where Badal's taps are greater (8 and 6).

Thus, we are left with only one valid scenario:

• Badal tapped exactly 6 times.
• Clive tapped exactly 8 times.

Let's review the final tally for all individuals:

• Alia = 6 taps
• Badal = 6 taps
• Clive = 8 taps
• Dilshan = 11 taps
• Ehsaan = 9 taps

As we can clearly see, Alia and Badal are the two people who tapped an equal number of times in total (6 times each). This proves why Option 4 is the correct choice.