Credit Card Numbers ^11 Each customer of Mobil Credit...

Credit Card Numbers $^{11}$ Each customer of Mobil Credit Corporation is given a 9 -digit number for computer identification purposes. a. If each digit can be any number from 0 to 9 , are there enough different account numbers for 10 million creditcard holders? b. Would there be if the digits were only 0 or $1 ?$

Credit Card Numbers $^{11}$ Each customer of Mobil Credit Corporation is given a 9 -digit number for computer identification purposes.
a. If each digit can be any number from 0 to 9 , are there enough different account numbers for 10 million creditcard holders?
b. Would there be if the digits were only 0 or $1 ?$

Key Concepts

Fundamental Counting Principle

This principle states that if a sequence of choices is made, where each choice is independent and has a certain number of possible outcomes, then the total number of outcomes is the product of the possibilities for each choice. In the context of assigning digit-based identification numbers, each digit’s possible values multiply together to give the total number of distinct numbers available.

Permutations with Replacement

When each digit in a number can be chosen from the same set and repetition is allowed, the total number of combinations is calculated by raising the number of choices per digit to the power of the number of digit positions. This concept is essential for determining the feasibility of generating enough unique account numbers from a fixed-digit format.

Base Number Systems

A base system defines the number of unique digits available for representing numbers. For instance, in a base-10 system, there are 10 digits (0 through 9) while in a base-2 system, there are only 2 digits (0 and 1). Understanding base systems helps in understanding how the count of possible combinations changes with different sets of digit values.

Comparison of Combinatorial Possibilities to Requirements

This concept involves comparing the total number of unique possible combinations, calculated using combinatorial methods, with the required number of unique identifiers (such as credit card numbers for customers). It is a critical step in determining whether a system can meet the demand for the required number of distinct identifiers.

Transcript

Please give Ace some feedback