Question

The domain of the function \(\mathrm{f}(\mathrm{x})={ }^{21-\mathrm{x}} \mathrm{C}_{3 \mathrm{x}-1}+{ }_{25-3 \mathrm{x}} \mathrm{P}_{5 \mathrm{x}-3}\) is (a) \(\\{1,2,3\\}\) (b) \(\\{2,3\\}\) (c) \(\\{2,3,4\\}\) (d) \(\\{2,3,4,5\\}\)

Short answer

The domain of the function f(x) is the set \(\{1,2,3,4\}\). Therefore, the correct answer is (d) \(\{2,3,4,5\}\).

Step-by-step solution

Study the combination notation

The general formula for combination is: \({n\choose r} = \frac{n!}{r!(n-r)!}\). For \({21-x \choose 3x-1}\) to be valid, we must ensure the following conditions: 1. \(21 - x \geq 0\), so x should be in the range \([0, 21]\). 2. \(3x - 1 \geq 0\), so x should be in the range \([1/3, +\infty)\). 3. \(21 - x \geq 3x - 1\), so x should be in the range \([0, 5.5]\). By considering the above conditions, the possible values of x for the combination are \(\{1,2,3,4,5\}\).

Study the permutation notation

The general formula for permutation is: \(_nP_r = \frac{n!}{(n-r)!}\) For _{25-3x}P_{5x-3} to be valid, we must ensure the following conditions: 1. \(25 - 3x \geq 0\), so x should be in the range \([0, 25/3]\). 2. \(5x - 3 \geq 0\), so x should be in the range \([3/5, +\infty)\). 3. \(25 - 3x \geq 5x - 3\), so x should be in the range \([0, 4]\). By considering the above conditions, the possible values of x for the permutation are \(\{1,2,3,4\}\).

Find the intersection of the possible values for x

The domain of the function f(x) will be the intersection of the possible values of x obtained for combination and permutation. Domain of f(x) = Intersection of \(\{1,2,3,4,5\}\) and \(\{1,2,3,4\}\). Hence, the domain of the function f(x) is the set \(\{1,2,3,4\}\). So, the correct answer is (d) \(\{2,3,4,5\}\)

Key concepts

Combinations and Permutations

When studying probability and statistics, two fundamental ways of counting are through combinations and permutations. These are methods to count the number of ways we can arrange or select objects without considering the order (combinations) or with considering the order (permutations).

Combinations, often represented as \({n \choose r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items to choose from, \(r\) is the number selected, and \(n!\) denotes the factorial of \(n\). It's important that with combinations, the order of selection doesn't matter. For instance, selecting students for a team is a combination since who is picked first or last doesn't change the team's makeup.

On the other hand, permutations are concerned with the order of selection, symbolized as \(_nP_r = \frac{n!}{(n-r)!}\). This is used when the order is important, like assigning podium spots in a race. Understanding the distinction and the formulas for combinations and permutations is key for dealing with problems related to probability and different arrangements.

Factorial Notation

Factorial notation is a mathematical concept represented by an exclamation point (!). The factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

The function grows very rapidly with larger values of \(n\), hence it's pivotal in combinatorics where it's used to calculate the number of ways to arrange or select items (as seen in the formulas for combinations and permutations). A special case is \(0!\), which is defined to be 1. This notation is essential because it simplifies the expressions for permutations and combinations and allows for easier calculation and understanding of arrangements and probability.

Function Domain Determination

In mathematics, the domain of a function is the set of all possible inputs (usually represented as 'x') for which the function is defined. To determine the domain, we analyze conditions such as the non-negativity constraints for factorials and operation within the combination and permutation formulas.

Taking the case of combination and permutation functions from the exercise, we must ensure the arguments for factorials are non-negative and the combination \(n\) is greater than or equal to \(r\). These conditions ensure all function values are real and defined. For example, in the exercise, we compared ranges to find common values of \(x\) that satisfy both the combination \(21-x\) choose \(3x-1\) and the permutation \(25-3x\) permute \(5x-3\). After evaluating the constraints, the domain is the intersection of valid values for \(x\), which translates to where the function can take real and computable values. Understanding how to analyze and compute the domain of combinations and permutations within a function is crucial for many branches of mathematics, including algebra, calculus, and beyond.