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Chapter 10: Problem 4

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

Short Answer

Expert verified
The value of the binomial coefficient \(\left(\begin{array}{c}11 \ 1\end{array}\right)\) is 11.

Step by step solution

01

Identify n and k in the binomial coefficient

The given expression is \(\left(\begin{array}{c}11 \ 1\end{array}\right)\). Here, n=11 and k=1.
02

Apply Binomial Coefficient Formula

The formula for calculating binomial coefficient is \(\left(\begin{array}{c}n \ k\end{array}\right) = \frac{n!}{k!(n-k)!}\). Substituting the values of n and k into the formula, we get \[\frac{11!}{1!(11-1)!} = \frac{11!}{1! \cdot 10!}\]
03

Calculate Factorials

11! means multiplying all positive whole numbers from 1 to 11, while 1! and 10! represents multiplying all numbers from 1 to 1 and 1 to 10 respectively. We calculate this as: \(11! = 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1, 1! = 1 \text{ and } 10! = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\)
04

Simplify the Expression

The 10 factorial in the numerator and the denominator will cancel out. Thus, the expression becomes: \(\frac{11}{1} = 11\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Factorials
Factorials are a fundamental concept in mathematics, symbolized by the exclamation mark (!). The factorial of a number, say \( n \), denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Some important points about factorials are:
  • It鈥檚 used extensively in permutations and combinations.
  • \( 0! = 1 \) as a special case by definition.
  • Factorials grow very quickly with larger numbers.
In the context of binomial coefficients, factorials help in calculating combinations, which are ways to pick items from a larger pool.
Combinatorics and Binomial Coefficients
Combinatorics is the branch of mathematics dealing with combinations and permutations of objects. A key part of combinatorics is understanding binomial coefficients, used to find combinations without considering the order.

The binomial coefficient is denoted by \( \binom{n}{k} \) and calculated as \( \frac{n!}{k!(n-k)!} \). Here's how it works:
  • \( n \) represents the total number of items.
  • \( k \) is the number of items to choose.
  • The formula accounts for all possible combinations.
In our example, \( \binom{11}{1} \), it shows how to select 1 item from 11.

The result, calculated through factorials, is 11, as only one choice from many results in many solutions.
Exploring Pascal's Triangle
Pascal's Triangle is a geometric arrangement of binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it.

Here's how it connects to binomial coefficients:
  • The nth row corresponds to the coefficients in the expansion of \( (a+b)^{n} \).
  • The entries are the binomial coefficients \( \binom{n}{k} \).
  • It provides a visual way to compute combinations quickly.
For the example given, \( \binom{11}{1} \), you would find it in the second position of the 11th row of Pascal's Triangle. It visually represents how we choose 1 item from 11, reinforcing the calculated value of 11.

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Most popular questions from this chapter

In Exercises \(49-52,\) a single die is rolled twice. Find the probability of rolling. If you toss a fair coin seven times, what is the probability of getting all tails?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\).

Use mathematical induction to prove that each statement is true for every positive integer \(n\). $$\sum_{i=1}^{n} 5 \cdot 6^{i}=6\left(6^{n}-1\right)$$

Use mathematical induction to prove that each statement is true for every positive integer \(n\). 2 is a factor of \(n^{2}+3 n\)

Use mathematical induction to prove that each statement is true for every positive integer \(n\). $$\sum_{i=1}^{n} 7 \cdot 8^{i}=8\left(8^{n}-1\right)$$
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