Problem 2 - Algebra and Trigonometry

Question

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l} {7} \\ {2} \end{array}\right) $$

Step-by-Step Solution

Answer

The binomial coefficient \(\left(\begin{array}{l} 7 \ 2 \end{array}\right)\) equals 21.

Step 1 - Identify n and k values

In the given coefficient, n is the top number and k is the bottom number. Hence, the n-value is 7 and the k-value is 2.

Step 2 - Evaluate factorials

Calculate the factorials of both n and k. \(!7 = 7*6*5*4*3*2*1 = 5040\) and \(!2 = 2*1 = 2\). It's also necessary to calculate \((n-k)!\), hence, \(!5 = 5*4*3*2*1 = 120\).

Step 3 - Substitute into formula and simplify

Substitute 7 for n, 2 for k, 5040 for 7!, 2 for 2! and 120 for 5! in the formula, we have \(\frac{5040}{2*120} = 21\). The answe is 21.

Key Concepts

Understanding Factorials

A factorial, represented by an exclamation mark (!), is a function that multiplies a given number by every whole number below it. For example, when we say 7 factorial (written as \( !7 \)), we mean the product of 7, 6, 5, 4, 3, 2, and 1, which equals 5040. Factorials are fundamental in various mathematical concepts, particularly in combinatorics, probability, and algebra.

Understanding how to compute a factorial is crucial when working with binomial coefficients, as we need to evaluate the factorials of both the top number (n) and the bottom number (k), as well as their difference (n-k). In combinatorial problems, factorials count the number of ways to arrange items, which links directly to the binomial coefficient calculations as seen in our example.

The Role of Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and combining groups of objects. The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose k objects from a larger set of n distinct objects, without regard to the order in which they are selected.

For our exemplar binomial coefficient \( \binom{7}{2} \), we are essentially looking to find out how many unique pairs we can select from a set of seven items. The binomial coefficient formula is expressed using factorials, which we've already covered, to facilitate these calculations: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \.
\ \]In this case, you'll notice that after calculating the necessary factorials, we then apply them within this formula to find our answer.

Probability and Binomial Coefficients

Probability is the measure of the likelihood that an event will occur and is quantified as a number between 0 and 1. Binomial coefficients play a critical role in probability, especially in scenarios where we deal with binary outcomes, like flipping a coin or answering true or false questions.

In such cases, a binomial coefficient can determine the number of different ways an event with two outcomes can occur a certain number of times within a given number of trials. Understanding how to calculate binomial coefficients is thus fundamentally important for solving probability problems that follow a binomial distribution. For students, recognizing the relation between factorials, combinatorics, and probability enables a comprehensive grasp of how to approach and solve these types of mathematical problems.