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Chapter 8: Problem 20

What is the generating function for the sequence \(\left\\{c_{k}\right\\}\) where \(c_{k}\) represents the number of ways to make change for \(k\) pesos using bills worth 10 pesos, 20 pesos, 50 pesos, and 100 pesos?

Short Answer

Expert verified
The generating function is \( \frac{1}{(1 - x^{10})(1 - x^{20})(1 - x^{50})(1 - x^{100})} \).

Step by step solution

01

Identify the problem

The task is to find the generating function for a given sequence where each term represents the number of ways to make change for a specified amount using specific denominations.
02

Understand generating functions

Generating functions are formal power series where the coefficient of each term corresponds to the sequence value. For example, a generating function for a sequence \{a_n\} would be given by \(G(a_n) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots\).
03

List available denominations

The bills available are 10 pesos, 20 pesos, 50 pesos, and 100 pesos.
04

Set up individual generating functions

For each bill denomination, create a generating function. Each bill denomination contributes to the generating function as follows: \((1 + x^{10} + x^{20} + x^{30} + \ldots)\) for 10 pesos, \((1 + x^{20} + x^{40} + x^{60} + \ldots)\) for 20 pesos, \((1 + x^{50} + x^{100} + x^{150} + \ldots)\) for 50 pesos, and \((1 + x^{100} + x^{200} + x^{300} + \ldots)\) for 100 pesos.
05

Write the generating function for each denomination as a series

Recognize that each series is a geometric series and can be written as \(\sum_{i=0}^{\infty} x^{n*i} = \frac{1}{1 - x^n}\) where \(n\) is the denomination. Therefore, the generating functions for the denominations are: \(\frac{1}{1 - x^{10}}\), \(\frac{1}{1 - x^{20}}\), \(\frac{1}{1 - x^{50}}\), and \(\frac{1}{1 - x^{100}}\).
06

Combine the generating functions

To find the overall generating function, multiply the individual generating functions: \[ G(x) = \frac{1}{1 - x^{10}} \cdot \frac{1}{1 - x^{20}} \cdot \frac{1}{1 - x^{50}} \cdot \frac{1}{1 - x^{100}} \]

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Key Concepts

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

sequence
A sequence, in mathematics, is an ordered list of numbers, symbols, or objects. When discussing sequences, a common notation used is \(\{a_n\}\), where \(a_n\) represents the term in the sequence. Sequences can be finite or infinite.

In the context of the given problem, \(c_k\) is a sequence where each \(c_k\) represents the number of ways to make change for \(k\) pesos using bills of different denominations. Understanding sequences is essential because each term in the sequence has a specific role and position, making patterns easier to analyze and solve.
geometric series
A geometric series is a series with a constant ratio between successive terms. It is written as:
\[a + ar + ar^2 + ar^3 + \ldots\],

where \(a\) is the first term and \(r\) is the common ratio.

For example, the series \(1 + x^n + x^{2n} + x^{3n} + \ldots\) is a geometric series with a ratio of \(x^n\). A key property of geometric series is that if the absolute value of \(x\) is less than 1, the series converges to:
\[\frac{1}{1-x}\]

In our problem, we formulated generating functions for each bill denomination as geometric series:
\[\frac{1}{1 - x^{10}}\] for 10 pesos,

\[\frac{1}{1 - x^{20}}\] for 20 pesos,

\[\frac{1}{1 - x^{50}}\] for 50 pesos,

\[\frac{1}{1 - x^{100}}\] for 100 pesos.
denominations
Denominations refer to the face value of a currency bill or coin. Understanding the concept of denominations is essential in problems that deal with making change or counting money, such as the given exercise.

In our problem, we need to consider how many ways we can use bills of 10 pesos, 20 pesos, 50 pesos, and 100 pesos to sum up to a total of \(k\) pesos. Each bill denomination is represented by a geometric series generating function:
  • 10 pesos: \[(1 + x^{10} + x^{20} + x^{30} + \ldots) \] -- written as \[\frac{1}{1 - x^{10}}\]

  • 20 pesos: \[(1 + x^{20} + x^{40} + x^{60} + \ldots)\] -- written as \[\frac{1}{1 - x^{20}}\]

  • 50 pesos: \[(1 + x^{50} + x^{100} + x^{150} + \ldots)\] -- written as \[\frac{1}{1 - x^{50}}\]

  • 100 pesos: \[(1 + x^{100} + x^{200} + x^{300} + \ldots)\] -- written as \[\frac{1}{1 - x^{100}}\]


By combining these generating functions, we obtain the overall generating function to determine the number of ways to make change for \(k\) pesos using the given denominations:
\[ G(x) = \frac{1}{1 - x^{10}} \cdot \frac{1}{1 - x^{20}} \cdot \frac{1}{1 - x^{50}} \cdot \frac{1}{1 - x^{100}} \].

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Most popular questions from this chapter

a) Show that if \(n\) is a positive integer, then $$ \left(\begin{array}{c}{-1 / 2} \\\ {n}\end{array}\right)=\left(\begin{array}{c}{2 n} \\ {n}\end{array}\right) /(-4)^{n} $$ b) Use the extended binomial theorem and part (a) to show that the coefficient of \(x^{n}\) in the expansion of \((1-4 x)^{-1 / 2}\) is \(\left(\begin{array}{c}{2 n} \\ {n}\end{array}\right)\) for all nonnegative integers \(n .\)

a) Find a recurrence relation for the number of ternary strings of length n that contain either two consecutive 0s or two consecutive 1s. b) What are the initial conditions? c) How many ternary strings of length six contain two consecutive 0s or two consecutive 1s?

Messages are transmitted over a communications channel using two signals. The transmittal of one signal requires 1 microsecond, and the transmittal of the other signal requires 2 microseconds. a) Find a recurrence relation for the number of different messages consisting of sequences of these two signals, where each signal in the message is immediately followed by the next signal, that can be sent in n microseconds. b) What are the initial conditions? c) How many different messages can be sent in 10 microseconds using these two signals?

Use the principle of inclusion-exclusion to derive a formula for \(\phi(n)\) when the prime factorization of \(n\) is $$n=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{m}^{a_{m}}$$

In the Tower of Hanoi puzzle, suppose our goal is to transfer all \(n\) disks from peg 1 to peg \(3,\) but we cannot move a disk directly between pegs 1 and \(3 .\) Each move of a disk must be a move involving peg \(2 .\) As usual, we cannot place a disk on top of a smaller disk. a) Find a recurrence relation for the number of moves required to solve the puzzle for \(n\) disks with this added restriction. b) Solve this recurrence relation to find a formula for the number of moves required to solve the puzzle for \(n\) disks. c) How many different arrangements are there of the \(n\) disks on three pegs so that no disk is on top of a smaller disk? d) Show that every allowable arrangement of the \(n\) disks occurs in the solution of this variation of the puzzle.
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