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 \(\left(\begin{array}{l}k \\ 0\end{array}\right)=1\). $$ \left(\begin{array}{c} -2 \\ 2 \end{array}\right) $$

Short answer

3

Step-by-step solution

Understand the formula

The presented formula \( \left(\begin{array}{l} k \ n \end{array}\right) \) is a standard form for binomial coefficients which are often found in expansion of binomial expressions. It can be calculated as the product of 'k', '(k-1)', '(k-2)', '...', until '(k-n+1)' and then divided by 'n' factorial.

Substitute the given values into the formula

Now we need to substitute the given values into the formula. Here, 'k' is -2 and 'n' is 2. So, the formula becomes \( \frac{(-2)(-2-1)}{2!} \)

Simplify the formula and calculate

Next we simplify the numerator and denominator separately. Numerator becomes \((-2) * (-3) = 6\). The denominator is calculated as '2!' which equals to \('2 * 1 = 2'\). So now the formula is \( \frac{6}{2} \)

Evaluate the final expression

On simply evaluating this expression \( \frac{6}{2} \), we get the answer as '3'

Key concepts

Calculus

Calculus, the mathematical study of continuous change, is a branch that deals with limits, functions, derivatives, integrals, and infinite series. While it may seem unrelated at first glance, calculus is actually related to binomial coefficients when we investigate functions that have binomial expressions and need to be differentiated or integrated.

Understanding binomial coefficients can be beneficial in calculus when dealing with Taylor or Maclaurin series, where we need to expand functions into polynomials. In this context, the binomial coefficients tell us the weight of each term in the expansion. Simplifying binomial coefficients through factorial notation or by recognizing patterns can thus be a crucial skill when taking calculus to higher levels of expanding and solving complex functions.

Factorial Notation

Factorial notation is a mathematical expression for the product of a natural number and all the natural numbers below it, denoted by an exclamation mark (!). For example, the factorial of 4, written as '4!', is calculated as
\( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Application in the Binomial Coefficient

The binomial coefficient formula utilizes factorial notation to calculate the number of ways to choose 'n' elements out of a set of 'k' elements, without regard to the order. It serves to prevent redundancy in counting and to simplify vast quantities into manageable expressions. Applying factorial notation correctly is crucial, as even a single error in calculating a factorial can lead to an incorrect binomial coefficient.

Binomial Theorem

The binomial theorem provides a quick way to expand binomial expressions raised to any power. It states that the expression \( (a + b)^n \) can be expanded into the sum of terms in the form of \( a^k b^{n-k} \), where the coefficient of each term is a specific binomial coefficient given by the formula \( \left(\begin{array}{l} n \ k \end{array}\right) = \frac{n!}{k!(n-k)!} \).

These coefficients, termed as binomial coefficients, are essential because they determine the weights of respective terms in the expansion.

Connection to Calculus

The theorem aligns with calculus in the sense that both involve the concept of approximation. In calculus, we approximate functions using their derivatives, while in the binomial theorem, we can approximate large binomials to a certain degree, which is an essential aspect of probability, statistics, and various calculus applications.