Question

This problem concerns functions \(f:\\{1,2,3,4,5,6,7\\} \rightarrow\\{0,1,2,3,4\\} .\) How many such functions have the property that \(\left|f^{-1}(\\{3\\})\right|=3\) ?

Short answer

The total number of desired functions is \( _7C_3 \) * \( 5^4 \). In numerical terms, this amounts to 35*625=21875.

Step-by-step solution

Select Three Elements

First, select 3 elements out of the 7 from the domain that will map to 3 in the codomain. This can be done in \( _7C_3 \) ways. The notation \( _nC_r \) denotes the number of ways to choose r elements from a set of n elements without considering the order. This is also known as binomial coefficient.

Distribute Remaining Elements

The remaining 4 elements can map to any of the 5 elements in the codomain, including 3. The total number of ways to do this is \( 5^4 \) by the product rule (4 elements, where each has 5 possible mappings)

Compute Total Functions

Finally, the total number of desired functions is the product of the number of ways to choose 3 elements and the number of ways to distribute the remaining 4, which is \( _7C_3 \) * \( 5^4 \).

Key concepts

Binomial Coefficient

The binomial coefficient is a key concept in combinatorics, expressed as \(_nC_r\) or \(\binom{n}{r}\), representing the number of ways to choose a subset of \(r\) elements from a larger set of \(n\) distinct elements without regard to the order of selection.

The mathematics behind it is grounded in the formula:\[\binom{n}{r} = \frac{n!}{r! \cdot (n - r)!}\],where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\). For example, choosing 3 elements from a set of 7, as shown in our exercise, involves calculating \(_7C_3\). This has practical applications not only in theoretical exercises but also in various fields such as probability, statistics, and even in constructing combinations in lotteries or tournament brackets.

Improving Conceptual Understanding

To better understand the binomial coefficient:

• Consider visualizing the selection process by drawing a set and picking elements one by one, noting that the order of selection does not change the outcome.
• Experiment with small values of \(n\) and \(r\) to compute the coefficients by hand, reinforcing the concept of factorial division.

Product Rule

The product rule in mathematics is a fundamental counting principle that determines the number of ways multiple independent choices can be made. When an event can occur in \(m\) ways and a subsequent event can occur in \(n\) ways, the total number of outcomes for both events is the product \(m \cdot n\).For example, if you have 4 positions to fill with 5 possible options for each, then the number of unique combinations is found by multiplying 5 by itself 4 times, expressed as \(5^4\).

In the context of our exercise, after selecting 3 elements to map to the number 3 in the codomain, we have to distribute the remaining 4 elements. Since each of these can map to any of the 5 elements in the codomain, we apply the product rule to calculate this as \(5^4\).

Applying the Product Rule

To grasp the product rule:

• Practice with scenarios involving step-by-step decision processes, noting how each choice leads to a new set of possibilities.
• Relate the product rule to real-life situations, like picking a meal with multiple options for the main course, side, and drink, and multiplying the options to find all possible meal combinations.

Domain and Codomain

In the realm of mathematics, particularly in functions, the domain and codomain are fundamental concepts that define the scope of a function. The domain is the set of all possible inputs for the function, often denoted as \(X\), while the codomain, denoted as \(Y\), represents the set of possible outputs.Consider the function \(f: X \rightarrow Y\); the domain here is \(X\) and the codomain is \(Y\). The actual outputs of the function, which are derived from the domain, form the range. It's important to note that the range is a subset of the codomain—the actual values that the function can produce, whereas the codomain is a set of all possible values that can come out of a function.

Exploring Domain and Codomain

To deepen understanding of this concept:

• Visualize functions as machines where inputs from the domain are processed to give outputs in the codomain.
• Create your own functions by defining a domain and codomain, then determine which elements map to each other to form the range of the function.

In the provided exercise, the domain includes the numbers 1 through 7, and the codomain includes the numbers 0 through 4. The function question asks specifically about the preimage of the number 3 in the codomain, which is the subset of the domain that maps to 3.