Problem 4 - Algebra and Trigonometry

Question

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l} {11} \\ {1} \end{array}\right) $$

Step-by-Step Solution

Answer

The binomial coefficient \( (11 1) = 11 \).

Step 1: Identifying Values

The given binomial coefficient is \( (11 1) \), therefore \( n = 11 \) and \( k = 1 \).

Step 2: Substituting into the Formula

Substitute the values of n and k into the formula for the binomial coefficient. This gives \( 11C1 = 11!/(1!(11-1)!) \).

Step 3: Calculating Factorials

Proceed by calculating the factorials. The factorial of a number is the product of all positive integers less than or equal to that number. Using this definition, \( 11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 \), \( 1! = 1 \), and \( (11-1)! = 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 \).

Step 4: Simplifying

After cancellation, we get \( 11C1 = 11 \). So the binomial coefficient for choosing 1 item from a set of 11 is 11.

Key Concepts

Factorials

Factorials are a key concept in mathematics and are especially significant when dealing with permutations and combinations. A factorial is denoted by an exclamation mark (!), and it's the product of all positive integers up to a given number. For example:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• \(0! = 1\) by definition, which is important in combinatorics and simplifies various formulas.

Factorials grow very quickly with larger numbers, which makes them a useful tool in combinatorics for counting arrangements and selections. When calculating a binomial coefficient like \(inom{11}{1}\), we utilize factorials to determine the total possible arrangements or selections within a given set.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It's essential in statistics, probability, and computer science for analyzing choices and permutations. A binomial coefficient is a combinatorial tool that tells us how many ways we can pick a set of elements from a larger group. The binomial coefficient for "n choose k," written as \(inom{n}{k}\), represents the number of combinations of n items taken k at a time.

• Formula: \(inom{n}{k} = \frac{n!}{k!(n-k)!}\)

In our example, \(inom{11}{1}\), we're choosing 1 item out of 11, which is straightforward because there are 11 possible choices.

Algebra

Algebra involves manipulating mathematical symbols and formulas to solve equations or describe mathematical relationships. In the context of binomial coefficients, algebra simplifies the arrangement of items into understandable formulas. It aids in substituting values and simplifying expressions like factorial representations. For instance, using the combination formula \(inom{n}{k} = \frac{n!}{k!(n-k)!}\), we can substitute values for \(n\) and \(k\). In our case, \(\binom{11}{1}\) becomes \(\frac{11!}{1!\cdot10!}\). Through algebraic simplification, unnecessary terms are canceled, such as the factorials 10!, leading directly to the result.

Binomial Theorem

The binomial theorem is a fundamental result in algebra that expresses the expansion of powers of a binomial. It provides a formula for expanding expressions of the form \((x + y)^n\). The coefficients of the expanded terms are given by binomial coefficients such as \(\binom{n}{k}\). So, the binomial theorem states:\((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)This theorem is crucial in calculus and probability theory for series expansions and calculations.In simpler terms, each term of the expansion corresponds to a combination of choosing items \(k\) at a time from \(n\) items. For instance, in the exercise with \(\binom{11}{1}\), it relates to choosing a single term when evaluating the binomial expansion.