Earlier when I attended a close door meeting to discuss the outsourcing of CAT papers. Including the chairman there were n people viz. A, B, C, D, E, F,… etc. AS per the convention everyone got some chocolates in the following manner. As A,B,C,D,E,… etc. received 1, 2, 3, 4, 5, ... etc. chocolates respectively. Before anyone had eaten a bit of chocolate, due to some urgent call, the chairmam left the meeting with his chocolates. Later on the rest attendants recollected their chocolates in a box and then redistributed all the chocolates evenly among themselves and thus everyone received 13 chocolates. : Who is the chairman of meeting?

Options

A. 1. A
B. 2. Q
C. 3. M
D. 4. can't be determined

Answer

To solve the problem, we need to analyze the distribution of chocolates and how it relates to the number of people present in the meeting, including the chairman.

### Step-by-Step Solution:

1. **Understanding the Distribution of Chocolates**:
Each person (A, B, C, D, E, etc.) receives chocolates in an increasing order:
- A receives 1 chocolate
- B receives 2 chocolates
- C receives 3 chocolates
- D receives 4 chocolates
- E receives 5 chocolates
- And so on...

If there are n people, the n-th person (let's say person N) receives n chocolates.

2. **Total Chocolates Before the Chairman Leaves**:
The total number of chocolates given to all n people can be calculated using the formula for the sum of the first n natural numbers:
\[ \text{Total Chocolates} = 1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2} \]

3. **Chocolates After the Chairman Leaves**:
Let’s denote the chairman as person C (who received C chocolates). If the chairman leaves with his chocolates, the remaining chocolates are:
\[ \text{Remaining Chocolates} = \frac{n(n + 1)}{2} - C \]

4. **Redistribution of Chocolates**:
After the chairman leaves, the remaining n-1 people redistribute the chocolates evenly. According to the problem, each of these n-1 people receives 13 chocolates:
\[ \text{Chocolates per person} = \frac{\text{Remaining Chocolates}}{n - 1} = 13 \]
Therefore:
\[ \text{Remaining Chocolates} = 13(n - 1) \]

5. **Setting Up the Equation**:
Now we can set up the equation:
\[ \frac{n(n + 1)}{2} - C = 13(n - 1) \]

6. **Rearranging the Equation**:
Rearranging gives us:
\[ C = \frac{n(n + 1)}{2} - 13(n - 1) \]

7. **Simplifying**:
Simplifying the right side:
\[ C = \frac{n(n + 1)}{2} - 13n + 13 \]
\[ C = \frac{n(n + 1) - 26n + 26}{2} \]
\[ C = \frac{n^2 - 25n + 26}{2} \]

8. **Finding Possible Values of C**:
Since C must be a positive integer (as it represents the number of chocolates), we need to find values of n such that \( n^2 - 25n + 26 \) is even and positive.

9. **Testing Values of n**:
We can test integer values for n to find a valid C:
- For n = 26:
\[ C = \frac{26^2 - 25 \times 26 + 26}{2} = \frac{676 - 650 + 26}{2} = \frac{52}{2} = 26 \]
- For n = 25:
\[ C = \frac{25^2 - 25 \times 25 + 26}{2} = \frac{625 - 625 + 26}{2} = \frac{26}{2} = 13 \]
- For n = 24:
\[ C = \frac{24^2 - 25 \times 24 + 26}{2} = \frac{576 - 600 + 26}{2} = \frac{2}{2} = 1 \]

10. **Conclusion**:
Since there are multiple valid values for C based on the values of n tested, we cannot definitively determine who the chairman is without additional information about the total number of attendees.

### Final Answer:
The answer is that we cannot determine who the chairman is based on the information provided.