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. : Maximum how many chocolates were there to be received by all of them, initially?

Options

A. 300
B. 676
C. 351
D. none of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the distribution of chocolates
Each person from A to the chairman (let's denote the chairman as C) received chocolates equal to their position in the sequence. So, if there are \( n \) people, they received chocolates as follows:
- A: 1 chocolate
- B: 2 chocolates
- C: 3 chocolates
- ...
- N: \( n \) chocolates

Thus, the total number of chocolates received by all \( n \) people is:
\[ \text{Total Chocolates} = 1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2} \]

### Step 2: Account for the chairman leaving
The chairman (C) leaves with his chocolates. Therefore, the total number of chocolates left after he leaves is:
\[ \text{Chocolates left} = \frac{n(n + 1)}{2} - n = \frac{n(n + 1)}{2} - n = \frac{n(n + 1) - 2n}{2} = \frac{n^2 - n}{2} \]

### Step 3: Redistribute the chocolates
After the chairman leaves, the remaining \( n - 1 \) people redistribute the chocolates evenly. Each person receives 13 chocolates:
\[ \text{Chocolates per person} = \frac{\text{Chocolates left}}{n - 1} = 13 \]
Thus, we can set up the equation:
\[ \frac{n^2 - n}{2(n - 1)} = 13 \]

### Step 4: Solve for \( n \)
Multiplying both sides by \( 2(n - 1) \) gives:
\[ n^2 - n = 26(n - 1) \]
Expanding the right side:
\[ n^2 - n = 26n - 26 \]
Rearranging the equation:
\[ n^2 - 27n + 26 = 0 \]

### Step 5: Factor the quadratic equation
Now, we can factor the quadratic:
\[ (n - 1)(n - 26) = 0 \]
Thus, the solutions for \( n \) are:
\[ n = 1 \quad \text{or} \quad n = 26 \]

### Step 6: Determine the maximum number of chocolates
Since we are looking for the maximum number of chocolates, we take \( n = 26 \):
\[ \text{Total Chocolates} = \frac{26(26 + 1)}{2} = \frac{26 \times 27}{2} = 351 \]

### Conclusion
The maximum number of chocolates that were initially received by all of them is **351**.

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