Queue Problem Analysis

The problem provides information about the positions of three persons, A, B, and C, in a queue. We are given their positions relative to the front or last, and the number of people between two of them. We need to find the number of people between B and C.

Queue Positions

Let's break down the given information:

• A is 14th from the front.
• B is 17th from the last.
• C is exactly in the middle of the queue.
• There are 16 persons between C and A.

Determining C's Position

A is 14th from the front. Since C is in the middle and there are 16 persons between A and C, C must be further down the queue than A.

C's position from the front can be calculated as:

$\text{C's position from front} = \text{A's position from front} + \text{Number of persons between A and C} + 1$ (for C itself)

$\text{C's position from front} = 14 + 16 + 1 = 31$

So, C is 31st from the front of the queue.

Calculating Total Queue Size

Since C is exactly in the middle, the number of persons before C is equal to the number of persons after C.

Number of persons before C = C's position from front - 1

$\text{Number of persons before C} = 31 - 1 = 30$

Number of persons after C = Number of persons before C = 30

The total number of persons in the queue is the sum of persons before C, C himself, and persons after C.

$\text{Total persons} = (\text{Persons before C}) + 1 + (\text{Persons after C})$

$\text{Total persons} = 30 + 1 + 30 = 61$

There are 61 persons in the queue.

Finding B's Position from Front

B is 17th from the last in the queue of 61 persons.

B's position from the front can be calculated using the total number of persons and B's position from the last:

$\text{B's position from front} = (\text{Total persons}) - (\text{B's position from last}) + 1$

$\text{B's position from front} = 61 - 17 + 1$

$\text{B's position from front} = 44 + 1 = 45$

So, B is 45th from the front of the queue.

Persons Between B and C

We know C is 31st from the front and B is 45th from the front. Since B's position (45) is greater than C's position (31), B is behind C in the queue.

The number of persons between two individuals in a queue when their positions from the same end are known is calculated as:

$\text{Persons between} = (\text{Position of the person further from the end}) - (\text{Position of the person closer to the end}) - 1$

Using positions from the front:

$\text{Persons between B and C} = (\text{B's position from front}) - (\text{C's position from front}) - 1$

$\text{Persons between B and C} = 45 - 31 - 1$

$\text{Persons between B and C} = 14 - 1 = 13$

There are 13 persons between B and C.