Question

How many seven-digit sequences can be formed using the svmbols \([0,1,2,3] ?\)

Short answer

There are 16,384 seven-digit sequences possible.

Step-by-step solution

Understanding the Problem

We are given four symbols: \(0, 1, 2, 3\), and we need to form a seven-digit sequence using these symbols. Each digit of the sequence can be any of these four symbols.

Calculating the Total Number of Sequences

For each of the seven positions in the sequence, we have 4 possible choices (0, 1, 2, or 3). Hence, by the multiplication principle, the total number of sequences is \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

Expressing the Problem as an Exponential

The formula from the previous step can be simplified as an exponent: \(4^7\), because we have 7 positions and each has 4 options.

Calculating the Exponential Value

Calculate \(4^7\): \[4^7 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 16384\]

Key concepts

discrete mathematics

Discrete mathematics is the branch of mathematics dealing with discrete objects. Unlike continuous mathematics, which deals with topics like calculus and differential equations, discrete math focuses on topics that involve distinct, separate values. In our exercise, we deal with sequences that are an example of discrete structures. These sequences utilize a limited set of symbols (0, 1, 2, 3), making the problem finite and defined.
Working with sequences is a common task in discrete mathematics because it involves counting distinct configurations or arrangements. This leads to an exploration of combinatorics, which is a key component of discrete math. Combinatorics gives us tools like the multiplication principle and exponential counting to solve problems, just like finding the number of seven-digit sequences.

• Combinatorics: Managing counting and arrangement tasks.
• Finite Structures: Working with limited objects, like sequences.
• Sequences: Understanding arrangements of symbols.

multiplication principle

The multiplication principle is a fundamental concept in combinatorics and discrete mathematics. It allows us to determine the number of possible outcomes when there are multiple stages or events, each with a set number of possible outcomes. This principle operates on the idea that if there are "m" ways to do one thing and "n" ways to do another, there are \(m \times n\) ways to do both.
In the given exercise, we used the multiplication principle to calculate the total number of seven-digit sequences that could be formed using the symbols 0, 1, 2, and 3. Each digit in the sequence provides us with four choices. Since there are seven positions to fill, by applying the multiplication principle, the calculations are broken into:

• 4 choices for the first digit,
• 4 choices for the second digit,
• Continue this until the seventh digit.

Thus, the total number of sequences is computed by multiplying these choices: \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^7\). This results in precisely 16,384 unique sequences.

exponential counting

Exponential counting involves using exponents to succinctly represent the number of possible outcomes. This technique is a natural extension of the multiplication principle, especially when dealing with repetitive trials or stages.
In the problem we're examining, after determining that each of the 7 positions in the sequence can independently be one of 4 options, the multiplication of these repeated selections can be conveniently expressed as an exponential function: \(4^7\).

• The base "4" represents the number of options per position.
• The exponent "7" corresponds to the total number of positions in the sequence.

When you compute \(4^7\), you perform an exponential calculation, resulting in a total of 16,384 possible seven-digit sequences with the symbols 0, 1, 2, and 3. This use of exponential notation simplifies lengthy multiplication and makes it easier to understand large combinations of choices.