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Chapter 6: Problem 40

Social Security Numbers A Social Security Number is a sequence of nine digits. a. How many Social Security Numbers are possible? b. How many of them begin with either 023 or 003 ? c. How many Social Security Numbers are possible if no two adjacent digits are the same? (For example, 235-93-2345 is permitted, but not 126-67-8189.)

Short Answer

Expert verified
a. There are \(10^9\) possible Social Security Numbers. b. There are \(2 * 10^6\) SSNs starting with either 023 or 003. c. There are \(10 * 9^8\) SSNs with no two adjacent digits being the same.

Step by step solution

01

Finding the total number of possible SSNs

An SSN is a sequence of 9 digits, where each digit can be any number from 0 to 9. To find the total number of possible SSNs, we simply use the counting principle: there are 10 choices for each of the 9 digits, so there are \(10^9\) possible SSNs.
02

Finding the number of SSNs starting with 023 or 003

To find the number of SSNs starting with 023 or 003, we can think of the problem as two separate groups of SSNs, one for each starting sequence. For each group, we have the following choices for the digits: - First 3 digits: fixed (023 or 003) - Remaining 6 digits: any number from 0 to 9. Using the counting principle, we have \(2 * 10^6\) possible SSNs starting with either 023 or 003.
03

Finding the number of SSNs without any two adjacent digits being the same

For finding the number of SSNs without any two adjacent digits being the same, we consider each digit position individually and find the number of choices available. 1. First digit: any of the 10 digits (0-9). 2. Second digit: any digit except the first one (9 choices). 3. Third digit: any digit except the second one (9 choices). 4. Fourth digit: any digit except the third one (9 choices). 5. Fifth digit: any digit except the fourth one (9 choices). 6. Sixth digit: any digit except the fifth one (9 choices). 7. Seventh digit: any digit except the sixth one (9 choices). 8. Eighth digit: any digit except the seventh one (9 choices). 9. Ninth digit: any digit except the eighth one (9 choices). Now, apply the counting principle: the total number of SSNs without any two adjacent digits being the same is \(10 * 9^8\). So the answers to the exercise are: a. There are \(10^9\) possible Social Security Numbers. b. There are \(2 * 10^6\) SSNs starting with either 023 or 003. c. There are \(10 * 9^8\) SSNs with no two adjacent digits being the same.

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Key Concepts

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

Counting Principle
In mathematics, the counting principle is a fundamental concept used to determine the number of possible outcomes in a sequence of events. It's particularly useful when each event is independent, meaning the outcome of one event does not affect the others.
For instance, when we talk about Social Security Numbers (SSNs), we apply the counting principle to calculate the total possible combinations. Each SSN has nine digits, and each digit can be any number from 0 to 9, giving us 10 choices per digit.
  • The first digit can be one of 10 numbers.
  • The second digit also has 10 options, and this pattern continues for all 9 digits.
Therefore, the total number of possible SSNs is found by multiplying the number of choices for each digit: \(10^9\).
This construction illustrates how the counting principle is a powerful tool for understanding large datasets and solving combinatorial problems involving sequences.
Permutations and Combinations
Permutations and combinations are techniques used to find the number of possible arrangements or selections of a set of objects.
In combinatorics, permutations consider the order of elements, making them useful for problems where the sequence matters. Conversely, combinations are used when the order doesn't matter.
A permutation is applicable to our SSN example in scenarios like determining how many SSNs start with a fixed set of digits, such as 023 or 003.
In this case, once the first three digits are fixed, the remaining six digits are permuted among the 10 possible numbers, leading to \(2 * 10^6\) options by the counting principle.
  • Permutations highlight the importance of sequence by fixing specific positions in problems.
  • Combinations would be more beneficial in different contexts where order is irrelevant.
Understanding both concepts helps solve a variety of mathematical problems efficiently, providing clear methods for calculating possible arrangements or selections.
Mathematical Problem Solving
Mathematical problem solving is a crucial skill in tackling exercises involving multiple steps and constraints. It involves applying various strategies and mathematical concepts to arrive at a solution.
In the context of Social Security Numbers, problem solving was applied in determining how to generate SSNs without two adjacent digits being the same. This required a methodical approach:
  • First decide on the initial digit, with 10 options.
  • For each subsequent digit, ensure it doesn't repeat the previous one, leaving 9 choices for each of these positions.
The result is a combination of systematic counting and constraint satisfaction, resulting in \(10 * 9^8\) non-repetitive SSNs.
Problem solving in this case requires breaking down the task, using logical reasoning, and understanding how different mathematical principles like counting and permutations work together to resolve a complex problem.

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Most popular questions from this chapter

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