Problem 73
Question
Evaluate the binomial coefficient using the formula \(\left(\begin{array}{l}k \\\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n !}\) where \(k\) is a real number, \(n\) is a positive integer, and \(\left(\begin{array}{l}k \\ 0\end{array}\right)=1\). $$ \left(\begin{array}{c} 0.5 \\ 4 \end{array}\right) $$
Step-by-Step Solution
Verified Answer
\(\left(\begin{array}{l}0.5 \ 4\end{array}\right) = 0.2734375\)
1Step 1: Substitute values into the formula
Start by substituting the values 0.5 for \(k\) and 4 for \(n\) into the formula \(\left(\begin{array}{l}k \ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3)}{n!}\). So, \(\left(\begin{array}{l}0.5 \ 4\end{array}\right)=\frac{0.5(0.5-1)(0.5-2)(0.5-3)}{4!}\).
2Step 2: Simplify the numerator
Perform the operations in brackets first for the numerator to get \(-0.5 \times -1.5 \times -2.5 \times -3.5\). After multiplication, the numerator becomes 6.5625.
3Step 3: Calculate the denominator
Calculate 4 factorial, written as 4! It is the product of all positive integers up to 4. So, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
4Step 4: Divide the numerator by the denominator
Now, divide the numerator by the denominator to get the final answer, thus, \(\left(\begin{array}{l}0.5 \ 4\end{array}\right) = \frac{6.5625}{24}\).
Key Concepts
Factorial FunctionNumerical ComputationCombinatorics
Factorial Function
The factorial function is one of the fundamental operations in mathematics, particularly in the field of combinatorics. It's denoted by an exclamation mark (!) and refers to the product of all positive integers up to a given number.
For example, the factorial of 4, written as '4!', is calculated as:
It is important to note that the factorial function is defined only for non-negative integers, and by convention, the factorial of zero (0!) is equal to 1. Factorials are critical in calculating permutations and combinations because they represent the number of ways in which items can be arranged.
For example, the factorial of 4, written as '4!', is calculated as:
- 4! = 4 × 3 × 2 × 1 = 24
It is important to note that the factorial function is defined only for non-negative integers, and by convention, the factorial of zero (0!) is equal to 1. Factorials are critical in calculating permutations and combinations because they represent the number of ways in which items can be arranged.
Numerical Computation
Numerical computation involves the use of numerical methods to solve mathematical problems. In the context of the binomial coefficient, it's a process where we substitute specific values into a given formula and perform arithmetic operations — such as addition, subtraction, multiplication, and division — to arrive at a numerical answer.
Through numerical computation, one can solve the binomial coefficient for non-integer values of 'k', as shown in the exercise. Here one must carefully perform each step of the calculation, mindful of the order of operations and the management of positive and negative signs, ensuring an accurate result.
Through numerical computation, one can solve the binomial coefficient for non-integer values of 'k', as shown in the exercise. Here one must carefully perform each step of the calculation, mindful of the order of operations and the management of positive and negative signs, ensuring an accurate result.
Combinatorics
Combinatorics is the branch of mathematics dealing with the study of countable, discrete structures. It includes understanding and solving problems related to counting, arrangement, and combination of elements within a set according to specified rules.
One of the key tools in combinatorics is the binomial coefficient, typically written as
It represents the number of ways 'k' elements can be selected from a set of 'n' elements without regard to the order. The binomial coefficient is utilized in various probability and statistics problems, such as determining the coefficients in the expansion of binomials raised to a power (the Binomial Theorem). Understanding the factorial function helps comprehend the binomial coefficient, as the former is used in the denominator of the coefficient's formula.
One of the key tools in combinatorics is the binomial coefficient, typically written as
- \(\left(\begin{array}{l}k\ n\end{array}\right)\)
It represents the number of ways 'k' elements can be selected from a set of 'n' elements without regard to the order. The binomial coefficient is utilized in various probability and statistics problems, such as determining the coefficients in the expansion of binomials raised to a power (the Binomial Theorem). Understanding the factorial function helps comprehend the binomial coefficient, as the former is used in the denominator of the coefficient's formula.
Other exercises in this chapter
Problem 72
Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{\ln n}{n}\right)^{n} $$
View solution Problem 73
Let \(a_{n}=\frac{n+1}{n}\). Discuss the convergence of \(\left\\{a_{n}\right\\}\) and \(\sum_{n=1}^{\infty} a_{n}\).
View solution Problem 73
Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{n}{(\ln n)^{n}} $$
View solution Problem 74
Evaluate the binomial coefficient using the formula \(\left(\begin{array}{l}k \\\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n !}\) where \
View solution