Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 64
Prove that
\(\left(\begin{array}{c}n \\\
r+1\end{array}\right)=\frac{n-r}{r+1}\left(\begin{array}{l}n \\\
r\end{array}\right), \quad 0 \leq r
Short Answer
Expert verified
The given relationship between binomial coefficients holds true.
Step by step solution
01
Understand the Problem
We are asked to prove a relationship involving binomial coefficients: \( \binom{n}{r+1} = \frac{n-r}{r+1} \binom{n}{r} \), where \(0 \leq r < n\). This involves understanding the properties of binomial coefficients.
02
Recall the Binomial Coefficient Definition
The binomial coefficient \( \binom{n}{r} \) is defined as \( \frac{n!}{r!(n-r)!} \). Similarly, \( \binom{n}{r+1} \) is \( \frac{n!}{(r+1)!(n-r-1)!} \). We will use these definitions in our proof.
03
Simplify \( \frac{n-r}{r+1} \binom{n}{r} \)
Substitute \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) into the expression \( \frac{n-r}{r+1} \binom{n}{r} \). This gives us:\[ \frac{n-r}{r+1} \cdot \frac{n!}{r!(n-r)!} = \frac{n!(n-r)}{(r+1)! (n-r)!} \].
04
Simplify \( \binom{n}{r+1} \)
Express \( \binom{n}{r+1} \) using its formula \( \binom{n}{r+1} = \frac{n!}{(r+1)!(n-r-1)!} \).
05
Compare the Expressions
In step 3, we derived \( \frac{n!(n-r)}{(r+1)! (n-r)!} \) and our expression for \( \binom{n}{r+1} \) is \( \frac{n!}{(r+1)!(n-r-1)!} \). Simplifying the first expression:\[ \frac{n!(n-r)}{(r+1)!(n-r)!} = \frac{n!}{(r+1)!(n-r-1)!} \]This equality holds because the \( (n-r) \) factor in the numerator cancels with one \( (n-r) \) term in the denominator's factorial.
06
Conclude the Proof
Since we showed that the expression from Step 3 simplifies to the exact form of \( \binom{n}{r+1} \), we have proven that the given relationship is true.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combinatorics
Combinatorics is a fascinating branch of mathematics that deals with counting, arrangement, and combination of objects. It's particularly useful in solving problems related to the selection of items from a collection, where the order doesn't matter, such as in our binomial coefficient exercise.
- The study often involves calculations of permutations (order matters) and combinations (order doesn't matter).
- Understanding combinatorics is key to solving many real-world problems, from figuring out probabilities in games to optimizing resource allocation.
- In the context of our exercise, the binomial coefficient \( \binom{n}{r} \) is a fundamental concept of combinatorics. It represents the number of ways to choose \( r \) items from \( n \) without considering the order.
Factorial Notation
Factorial notation plays a crucial role in combinatorics, especially when dealing with binomial coefficients. The expression \( n! \) (read as 'n factorial') is the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
- Factorials grow very quickly, which is why they are used to represent the number of permutations or combinations.
- In our exercise, the binomial coefficient \( \binom{n}{r} \) is defined using factorial notation as \( \frac{n!}{r!(n-r)!} \).
- This representation simplifies the calculation of combinations and makes it easier to manipulate the expressions for proof, as is required in our step-by-step solution.
Mathematical Proof
A mathematical proof is a logical argument that establishes the truth of a mathematical statement. In mathematics, proofs are essential for demonstrating the correctness of a hypothesis or formula, as seen with our binomial coefficient exercise.
- Proofs use logical reasoning, starting from known facts and axioms, to show that a proposition is universally true.
- In our exercise, the proof involves demonstrating that the expression \( \frac{n-r}{r+1} \binom{n}{r} \) simplifies directly to \( \binom{n}{r+1} \).
- This involves using the definition of binomial coefficients and manipulating the algebraic expression through careful simplification.
