Question

Two squares are chosen at random on a chessboard, the probability that they have a side in common is : (a) \(\frac{3}{32}\) (b) \(\frac{1}{32}\) (c) \(1 / 18\) (d) none of these

Short answer

In this exercise, we are asked to find the probability of two squares chosen at random on a chessboard having a side in common. The total number of ways to pick two squares is 1984, and the number of ways two squares have a side in common is 224. Therefore, the probability is \(\frac{224}{1984} = \frac{7}{62}\), which is not one of the given options. So, the correct answer is (d) none of these.

Step-by-step solution

Determine the total number of ways to pick two squares

On a chessboard, there are 64 squares. We can choose any two squares out of these 64. Using combinations formula, the total number of ways to choose 2 squares from 64 is given by: \(\binom{64}{2} = \frac{64!}{62!2!} = \frac{64*63}{2} = 64 * 31 = 1984\)

Determine the number of ways two squares have a side in common

To find the number of ways two squares have a side in common, we will consider horizontal and vertical common sides separately. A square can have a maximum of 2 horizontal common sides (left and right) except for squares on the extreme left and right columns, which only have 1 horizontal common side. Since there are 8 rows each with 6 squares having 2 horizontal common sides and 2 squares with 1 horizontal common side, the total number of horizontal common sides is: \((8 * 6 * 2) + (8 * 2) = 112\) Similarly, a square can have a maximum of 2 vertical common sides (top and bottom) except for squares on the top and bottom rows, which only have 1 vertical common side. \((8 * 6 * 2) + (8 * 2) = 112\) Adding both horizontal and vertical common sides, we get the total number of ways two squares have a side in common as 224.

Calculate the probability

Now that we have the total number of ways to pick two squares (1984) and the number of ways two squares have a side in common (224), we can find the probability by dividing these two numbers: \(P(\text{side in common}) = \frac{\text{number of ways with common side}}{\text{total number of ways}} = \frac{224}{1984} = \frac{28}{248} = \frac{7}{62}\) Since \(\frac{7}{62}\) is not one of the given options, the correct answer is: (d) none of these

Key concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, among others.

In our chessboard problem, combinatorics comes into play when counting the total number of ways two squares can be chosen and also the number of combinations where two squares share a common side. This counting is fundamental in calculating probabilities and understanding the structure of the chessboard, which in itself is a finite geometric arrangement.

Key terms in combinatorics, like permutations and combinations, help distinguish between different scenarios with or without regard to the order of selection. In our context, combinations are used because the order of selecting the squares is not important.

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion.

When applied to a chessboard scenario, such as the probability of choosing squares with a common side, the theory provides a mathematical framework for quantifying the likelihood of various outcomes. It involves the ratio of the favorable outcomes to all possible outcomes. This scenario illustrates a basic principle of probability: the likelihood of an event is calculated by dividing the number of ways that event can occur by the total number of all possible events.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics that deal with the various ways in which objects from a set can be selected, generally without replacement, to form subsets. This area of mathematics helps us to systematically count without actually enumerating all possible scenarios.

A permutation of a set is an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word 'permutation' also refers to the act or process of changing the linear order of an ordered set.

Combinations, on the other hand, refers to the selection of all or part of the set without regard to the order of the elements. For our chessboard problem, the combination is used to count the ways we can select two squares, disregarding their order because whether we select square A first or square B first, the outcome is the same two squares chosen.