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Chapter 13: Problem 7

Fill in each blank with the correct response. For any nonnegative integer \(n\), the binomial coefficient \({ }_{n} C_{0}\) is equal to _____.

Short Answer

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Step by step solution

01

Understand the Binomial Coefficient Notation

The binomial coefficient is denoted by \({}_{n}C_{k}\) and represents the number of ways to choose \k\ objects from \ objects without regard to order.
02

Identify the Given Values

In this problem, \k\ is given as 0. We need to determine \({}_{n}C_{0}\) for any nonnegative integer \.
03

Recall the Binomial Coefficient Formula

The binomial coefficient \({}_{n}C_{k}\) is calculated using the formula: \[ {}_{n}C_{k} = \frac{n!}{k!(n-k)!} \]
04

Apply the Formula to Given Values

Substitute 0 for \k\ in the formula: \[ {}_{n}C_{0} = \frac{n!}{0!(n-0)!} = \frac{n!}{0!n!} \]
05

Simplify the Expression

Since \0! = 1\, the expression simplifies to: \[ {}_{n}C_{0} = \frac{n!}{1 \times n!} = 1 \]
06

Conclusion

\({}_{n}C_{0}\) is equal to 1 for any nonnegative integer \.

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Key Concepts

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

Binomial Theorem
The binomial theorem is a fundamental principle in algebra that helps us expand expressions of the form \((a + b)^n\). It states that:[ (a + b)^n = \sum_{k=0}^{n}\ {}_{n}C_{k} \a^{n-k}b^{k}\ ]Where \({}_{n}C_{k}\) is the binomial coefficient. This theorem simplifies the expansion process, avoiding long multiplication steps. For example, using the binomial theorem, we can expand \((x + y)^3\) as follows:
  • \[(x + y)^3 = {}_{3}C_{0} x^3 y^0 + {}_{3}C_{1} x^2 y^1 + {}_{3}C_{2} x^1 y^2 + {}_{3}C_{3} x^0 y^3\]
  • \[= 1 \times x^3 + 3 \times x^2y + 3 \times xy^2 + 1 \times y^3\]
  • \[= x^3 + 3x^2y + 3xy^2 + y^3\]
You can see how each term in the expansion is determined by the binomial coefficients and the respective powers of \a\ and \b\.
Combinatorics
Combinatorics is a branch of mathematics focusing on counting, arranging, and finding patterns. It's essential for understanding probabilities and configurations. One crucial concept in combinatorics is the binomial coefficient, denoted by \({}_{n}C_{k}\). This coefficient counts the number of ways to choose \k\ objects from \ objects regardless of order.
  • \[( {}_{n}C_{k} = \frac{n!}{k!(n-k)!}\ )\]
  • Choosing 0 objects: \({}_{n}C_{0}\) will always be 1 since there's only one way to choose nothing from a set.
  • Choosing all objects: \({}_{n}C_{n}\) is also 1, as there's only one way to choose all from a set.
Combinatorics assists in solving diverse problems, from simple counting to intricate arrangement possibilities.
Factorial
The factorial of a nonnegative integer \, denoted as !, is the product of all positive integers less than or equal to \. Mathematically,= n \cdot (n-1) \cdot (n-2) \cdots2 \cdot1. Factorials are crucial in calculating permutations and combinations within combinatorics.
  • 0! By definition, \0!\ equals 1. This is essential when simplifying binomial coefficients.
  • Using Factorials in Binomial Coefficients: For example, to find \({}_{5}C_{2}\), we calculate: \[ {}_{5}C_{2} = \frac{5!}{2!(5-2)!} = \frac{5 \cdot 4\cdot 3!}{2 \cdot 1 \cdot 3!} = \frac{20}{2} = 10 \]
Understanding factorials is vital for mastering topics in combinatorics and applying the binomial coefficient formula effectively.

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