Solving Professor Meeting Attendance Problem

This problem requires calculating the overlap in attendance between meetings, given specific attendance numbers and the constraint that each professor attended exactly two meetings.

Define Variables and Information

Let the number of professors attending:

• Mumbai and Delhi meetings only be MD.
• Mumbai and Chennai meetings only be MC.
• Delhi and Chennai meetings only be DC.

We are given the total attendance for each city:

• Total attending Mumbai ($|M|$): 21
• Total attending Delhi ($|D|$): 27
• Total attending Chennai ($|C|$): 30

Since each professor attended exactly two meetings, the total attendance for each city can be expressed as the sum of professors attending the pairs including that city:

• Mumbai attendance: $MD + MC = 21$
• Delhi attendance: $MD + DC = 27$
• Chennai attendance: $MC + DC = 30$

We need to find the number of professors who attended both Chennai and Delhi meetings, which is DC.

Calculate Delhi and Chennai Attendance (DC)

We have a system of three linear equations:

1. $MD + MC = 21$
2. $MD + DC = 27$
3. $MC + DC = 30$

Add all three equations together:

$(MD + MC) + (MD + DC) + (MC + DC) = 21 + 27 + 30$

Simplify the left side:

$2 \times MD + 2 \times MC + 2 \times DC = 78$

Factor out the 2:

$2 \times (MD + MC + DC) = 78$

Divide by 2 to find the total number of professors (sum of all pairs):

$MD + MC + DC = \frac{78}{2} = 39$

To find DC, substitute the value of $MD + MC$ from equation (1) into the total sum:

$21 + DC = 39$

Solve for DC:

$DC = 39 - 21$

$DC = 18$

Thus, 18 professors attended both the Chennai and Delhi meetings.