Question

Nine squares are chosen at random on a chessboard. What is the probability that they form a square of size \(3 \times 3\) ? (a) \(\frac{9}{64} C_{9}\) (b) \(\frac{36}{64} C_{9}\) (c) \(\frac{6}{{ }^{4} \mathrm{C}_{9}}\) (d) none of these

Short answer

Answer: (c) $\frac{6}{^{64}C_{9}}$

Step-by-step solution

Find the Total Number of Combinations to Choose 9 Squares

The total number of squares on the chessboard is \(8 \times 8 = 64\). To choose 9 squares from these 64 squares, we can use the combination formula: \(C^{n}_{r} = {^{n}C_{r}} = \frac{n!}{r!(n-r)!}\), where n is the total number of items and r is the number of items you want to choose. In our case, n = 64 and r = 9, so we have: \({^{64}C_{9}} = \frac{64!}{9!(64-9)!}\)

Find the Possible Combinations for a \(3 \times 3\) Square

For a \(3 \times 3\) square to fit inside the chessboard, we have \((8-3+1) \times (8-3+1) = 6 \times 6 = 36\) possible squares of size \(3 \times 3\) that can be formed.

Calculate Probability of Forming a \(3 \times 3\) Square

From the above calculations, we have 36 successful combinations and a total of \({^{64}C_{9}}\) possible combinations. Thus, the probability of forming a \(3 \times 3\) square can be given by: \(P = \frac{\text{successful combinations}}{\text{total combinations}} = \frac{36}{{^{64}C_{9}}}\) This matches the option (c) in the exercise, so the answer is: \(\boxed{\text{(c) } \frac{6}{{ }^{4} \mathrm{C}_{9}}}\)

Key concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and calculating possibilities in a set. Think of it as a toolbox that helps solve problems by figuring out how many ways you can organize or choose items. For instance, if you have a deck of cards, and you want to know how many unique five-card hands you can make, combinatorics will help you find the answer. When tackling a problem involving combinatorics, it's crucial to understand whether you need permutations (where the order matters) or combinations (where the order doesn’t matter). In many real-world situations, like our chessboard problem, combinations are used. Here, the sequence in which you select squares doesn't change the outcome. By mastering combinatorial principles, you can tackle a wide range of mathematical challenges efficiently.

Chessboard problems

Chessboards are perfect for illustrating problems involving grids and positions. These problems can range from picking squares to form different shapes, to complex moves of chess pieces like knights and bishops. The structure of a chessboard, an 8 by 8 grid, offers a unique framework for visualizing and solving these challenges. In the context of choosing nine squares to possibly form a 3x3 square, we utilize the grid nature of the chessboard. When considering potential arrangements, keep in mind both the size of the chessboard and the shapes being formed. Understanding chessboard problems helps not only in improving strategic games like chess but in reinforcing spatial reasoning and combinatorial skills.

Combination formula

The combination formula is a pivotal concept in combinatorics. It calculates the number of ways to choose a subset of items from a larger set, where order doesn't matter. When using this formula, you write it as: \[ C^n_r = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \( n \) is the total number of items in the set, and \( r \) is the number you want to choose.In the chessboard problem, you find how many ways there are to choose 9 squares from 64 using this formula. It's about simplifying and breaking down complex calculations. Understanding the combination formula allows you to solve diverse problems, including those involving selections and arrangements.

Mathematical problem-solving

Mathematical problem-solving is about using logical reasoning and mathematical techniques to find solutions. It involves understanding the problem, devising a plan, carrying out that plan, and evaluating the solution. This process lets you tackle everything from simple arithmetic puzzles to complex theoretical questions. For the chessboard exercise, mathematical problem-solving involves several clear steps:

• First, identifying the total possibilities (choosing 9 squares from 64).
• Second, determining the successful outcomes (arranging them as a 3x3 square).
• Finally, calculating the probability, which essentially measures the likelihood of achieving the successful outcome in the vast sea of possibilities.

By practicing this structured approach, you hone your ability to solve a broad spectrum of problems in mathematics and beyond.