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Chapter 1: Problem 12

Consider the \(n \times n\) array whose entry in the \(i\) th row, \(j\) th column is \(i+j-1\). What is the smallest product of \(n\) numbers from this array, with one coming from each row and one from each column?

Short Answer

Expert verified
The smallest product is \(n!\).

Step by step solution

01

Identify the array structure

Given an array where the entry in the \(i\) th row, \(j\) th column is \(i+j-1\), it means the value at position (i, j) is calculated by adding the row number and column number, then subtracting 1.
02

Understand the problem's requirement

The problem asks to find the smallest product of \(n\) numbers from the array where each number comes from a different row and a different column.
03

Simplify and analyze the entries

Each row in the array is a sequence \[i, i+1, ..., i+(n-1)\]. Essentially, for any row \(i\), you can choose \(j=i\) to get the smallest number, which is \(i\).
04

Identify the smallest elements

For the n numbers, the smallest ones from each row (considering they need to be from different columns) simply results in choosing \(1\) from the first row, \(2\) from the second row, and so on, up to \(n\) from the \(n\)th row. These are \[1, 2, ..., n\].
05

Calculate the product

The product of the elements \{1, 2, ..., n\}\ is simply \{1 \times 2 \times 3 \times \ldots\ \times n\}\, which equals \(n!\).
06

Write the solution

Thus, the smallest product of \(n\) such numbers, one from each row and from each column, corresponds to \(1 \times 2 \times 3 \times ... \times (n-1) \times n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorial Optimization
Combinatorial optimization involves selecting the most efficient solution from a finite set of possible solutions. In this problem, we need to choose one number from each row and column of an array to achieve the smallest product. This means finding the optimal combination of elements that leads to the minimum possible product. Understanding combinatorial optimization can help solve complex problems by breaking them down into simpler, manageable parts.

In this matrix problem, combinatorial optimization is evidenced by selecting the smallest element from each row's sequence, while ensuring none are from the same column, thereby minimizing the final product.
Array Manipulation
Array manipulation refers to the methods and techniques used to change, analyze, and understand array data. Arrays are powerful data structures that store items sequentially. In this problem, we work with an array whose elements are defined by an equation: \(i + j - 1\).

To find the smallest product by taking one element from each row and column, we must manipulate the array to identify the smallest possible entries. Specifically, for any row i, selecting the element where \(j = i\) minimizes the value, making the resultant numbers \[1, 2, ..., n\].

Mastering array manipulation can greatly assist in efficiently navigating and solving mathematical problems involving sequential data.
Matrix Operations
Matrix operations are mathematical procedures that involve the manipulation of matrices. In this exercise, understanding the organization and properties of a matrix helps us identify the smallest product. The given matrix's entries are determined by \(i + j - 1\), creating lines of increasing integers across the rows and columns.

The process involves:
  • Understanding how to traverse through the matrix efficiently
  • Selecting one element from each row and column
  • Recognizing patterns within the matrix entries
These steps help in isolating the smallest possible overall product by selecting \[1, 2, ..., n\] from respective rows.

Familiarity with matrix operations is crucial for solving more advanced mathematical and combinatorial problems.
Factorial Calculation
Factorial calculation is a fundamental mathematical concept where \(n! = n \times (n-1) \times ... \times 2 \times 1\). In this problem, the smallest product from the matrix equates to calculating the factorial of n (\(n!\)).

The solution involves selecting the sequence of integers from 1 to n, as these integers are the smallest from each row, arranged diagonally. Hence, their product is:
  • \1\, \2\, \3\, ..., n\
  • Resulting in \(n!\)
Calculating factorials is integral in combinatorial problems and many areas of mathematics, providing grouped multiplicative outcomes based on a sequence's length.

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Most popular questions from this chapter

The mayor of Wohascum Center has ten pairs of dress socks, ranging through ten shades of color from medium gray (1) to black (10). When he has worn all ten pairs, the socks are washed and dried together. Unfortunately, the light in the laundry room is very poor and all the socks look black there; thus, the socks get paired at random after they are removed from the drier. A pair of socks is unacceptable for wearing if the colors of the two socks differ by more than one shade. What is the probability that the socks will be paired in such a way that all ten pairs are acceptable?

a. Find all lines which are tangent to both of the parabolas $$ y=x^2 \quad \text { and } \quad y=-x^2+4 x-4 $$ b. Now suppose \(f(x)\) and \(g(x)\) are any two quadratic polynomials. Find geometric criteria that determine the number of lines tangent to both of the parabolas \(y=f(x)\) and \(y=g(x)\).

For \(x \geq 0\), let \(y=f(x)\) be continuously differentiable, with positive, increasing derivative. Consider the ratio between the distance from \((0, f(0))\) to \((x, f(x))\) along the curve \(y=f(x)\) (the arc length from 0 to \(x\) ) and the straight-line distance from \((0, f(0))\) to \((x, f(x))\). Must this ratio have a limit as \(x \rightarrow \infty\) ? If so, what is the limit?

Can there be a multiplicative \(n \times n\) magic square \((n>1)\) with entries \(1,2, \ldots, n^2\) ? That is, does there exist an integer \(n>1\) for which the numbers \(1,2, \ldots, n^2\) can be placed in a square so that the product of all the numbers in any row or column is always the same?

Fast Eddie needs to double his money; he can only do so by playing a certain win-lose game, in which the probability of winning is \(p\). However, he can play this game as many or as few times as he wishes, and in a particular game he can bet any desired fraction of his bankroll. The game pays even money (the odds are one-to-one). Assuming he follows an optimal strategy if one is available, what is the probability, as a function of \(p\), that Fast Eddie will succeed in doubling his money?
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