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Chapter 6: Problem 9

Using only the digits \(1,3,4\), and 7 , (a) how many two-digit numbers can be formed? (b) how many three-digit numbers can be formed? (c) how many two- or three-digit numbers can be formed?

Short Answer

Expert verified
(a) 12, (b) 24, (c) 36 two- or three-digit numbers.

Step by step solution

01

Understand the Problem

We need to find how many two-digit and three-digit numbers we can form using the digits 1, 3, 4, and 7 without repetition.
02

Calculate Two-Digit Numbers

For a two-digit number, we have two positions: tens and units. We can use any of the four digits (1, 3, 4, 7) in the tens position, and any of the remaining three digits in the units position since repetition is not allowed. Therefore, the number of two-digit numbers is:\[4 \times 3 = 12\]
03

Calculate Three-Digit Numbers

For a three-digit number, we have three positions: hundreds, tens, and units. We can use any of the four digits in the hundreds position, any of the remaining three digits in the tens position, and any of the remaining two digits in the units position. Thus, the number of three-digit numbers is:\[4 \times 3 \times 2 = 24\]
04

Calculate Total Two- or Three-Digit Numbers

Finally, to find the total number of two- or three-digit numbers that can be formed, we simply add the results from Step 1 and Step 2:\[12 + 24 = 36\]

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Key Concepts

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

Permutations
Permutations are a fundamental concept in combinatorics, focusing on the arrangement of elements in a specific order. When dealing with permutations, the order of selection is crucial, unlike combinations where the order does not matter. In our original exercise, we are asked to find sequences consisting of two-digit and three-digit numbers.
  • For the two-digit numbers, we place one of the available digits in the tens place and choose from the remaining digits for the units place.
  • For three-digit numbers, we place one digit in the hundreds place, then select a digit for the tens place from the remaining options, and finally pick for the units place from what's left.
This step-by-step selection without repetition perfectly illustrates the concept of permutations.
Counting Techniques
Counting techniques are essential tools in mathematics that help determine the number of possible arrangements or selections.
In the context of this exercise, we use a straightforward counting technique called the Fundamental Principle of Counting, which states:
"If one event can occur in 'm' ways and a second can occur independently of the first in 'n' ways, then the two events can occur in 'm x n' ways."
This rule applies here:
  • To form a two-digit number, choose one of the four digits for the tens place, and then one of the remaining three for the units.
  • To form a three-digit number, select a digit for the hundreds place, another from the remaining digits for the tens, and then one from the remaining two for the units.
Each step decreases the number of available choices, ensuring we're organizing without repetition.
Discrete Mathematics
Discrete mathematics involves studying distinct and separate values or objects, unlike continuous mathematics, which deals with smooth and uninterrupted data. It includes structures such as integers, graphs, and statements in logic鈥攚hich are foundational for computer science and combinatorics.
The exercise at hand draws heavily from discrete math principles by considering each digit placement as an individual step, leading to a finite set of outcomes.
  • The clear separation of choices (digits 1, 3, 4, 7) and their arrangement forms a classic discrete mathematical problem.
  • The task of arranging these digits in piles鈥攂e it for two-digit or three-digit combinations鈥攇ives insight into finite structures.
Discrete math lays the groundwork for more complex topics like algorithms, coding theory, and probability, by fostering logical reasoning and problem-solving skills.

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Most popular questions from this chapter

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