Question

There are 15 stations from New Delhi to Mumbai. How many first class tickets can be printed to travel from one station to any other station? (1) 210 (2) 105 (3) 240 (4) 135

Short answer

Answer: 105

Step-by-step solution

Define the problem

We are given 15 stations and need to find the number of first class tickets that can be printed for travel from one station to another. The important thing to note here is that a ticket cannot be generated for a station to itself, meaning that the origin and destination stations cannot be the same.

Calculate combinations

We can use the combination formula to calculate the number of ways to choose two stations out of 15. The combination formula is given by: \[C(n,k) = \frac{n!}{k!(n-k)!}\] where C(n,k) represents the number of possible combinations, n is the total number of stations, k is the number of stations we're choosing, and n! is the factorial of n.

Apply the formula

In our case, n = 15 and k = 2 since we need to select 2 stations out of 15. Therefore, applying the formula: \[C(15,2) = \frac{15!}{2!(15-2)!} = \frac{15!}{2!13!}\]

Calculate factorials

Now, we'll calculate the factorials for 15, 2, and 13, which are: \[15! = 15 \times 14 \times 13!\] \[2! = 2 \times 1 = 2\] We can substitute these factorials back into the formula: \[\frac{15!}{2!13!} = \frac{15 \times 14 \times 13!}{2 \times 1 \times 13!}\]

Simplify the expression

Now we can simplify the expression: \[\frac{15 \times 14 \times 13!}{2 \times 1 \times 13!} = \frac{15 \times 14}{2 \times 1}\] Dividing both sides by 2: \[15 \times 7 = 105\] So, there are 105 possible first class tickets that can be printed to travel from one station to another out of a total of 15 stations. The correct answer is (2) 105.

Key concepts

Understanding Factorials

In mathematics, a factorial is a crucial concept often symbolized by the exclamation point. For a positive integer n, the factorial (denoted as \( n! \)) is the product of all positive integers less than or equal to n. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are widely used in calculations involving permutations and combinations.

One key thing to remember is that \( 0! = 1 \) by definition. Factorials grow rapidly with increasing numbers, which is why they're extensively used in combinatorial calculations. In our original problem, factorials help in figuring out how many different ways a number of items (stations) can be arranged or selected, which is fundamental in counting tickets.

Permutations and Combinations

Permutations and combinations are crucial techniques in combinatorics used to determine the number of ways objects can be arranged or selected.

• Permutations: This refers to the arrangement of objects in a specific order. The order matters here. For example, arranging the letters A, B, and C in different ways such as ABC, ACB, BAC, etc.
• Combinations: Here, the focus is on the selection of objects without regard to the order. In the context of the exercise, it involves selecting two stations out of the available 15, where the sequence does not matter.

Combinations are calculated using the formula \( C(n,k) = \frac{n!}{k!(n-k)!} \). In the exercise, combinations were used to determine how many pairs of stations could be formed, leading to the conclusion of 105 possible ticket combinations.

Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. It deals with understanding the likelihood or chances of different outcomes. Although not directly used in the problem, probability theory often involves permutations and combinations.

By calculating the possible arrangements or selections of events, we apply principles of probability to predict or understand various outcomes. In the travel ticket context, knowing the possible routes helps in assessing likelihoods or choices a passenger might take, thus indirectly linking to probability theory.

Discrete Mathematics

Discrete mathematics involves the study of distinct and separable values, often including topics like graph theory, logic, set theory, and combinatorics. It deals with finite or countable processes and includes calculations that are precise, such as determining the combinations of stations.

In the exercise, discrete mathematics principles were applied in the count of station combinations without repetition. Using factorials and combination formulas, problems of this nature are solved efficiently.

Discrete mathematics plays a vital role in computer science and information theory, making it integral to the development of algorithms and solving logistical problems like ticket printing for train stations.