Question

Show that $$N=C(2 n, n)=\frac{(n+1)(n+2) \ldots(n+n)}{n !}<2^{2 n}<(2 n+1) N$$

Short answer

Answer: The inequality for the binomial coefficient \(C(2n, n)\) is $$ N<C(2 n, n)<2^{2 n}<(2n+1)N.$$

Step-by-step solution

Find the expression for \(C(2n, n)\)

Using the formula for binomial coefficients, we can write \(C(2n, n)\) as $$C(2n, n) = \frac{(2n)!}{n!n!}.$$ Then, we can rewrite the numerator as $$ (2n)! = (2n)(2n-1)\ldots(n+1)n! $$ Substituting this expression in the formula for \(C(2n, n)\), we get $$ N = C(2n, n) = \frac{(n+1)(n+2) \ldots(n+n)}{n!}.$$

Compare \(N\) with \(2^{2n}\) and \((2n+1)N\)

Now, let's compare \(N\) with \(2^{2n}\). We can observe that $$ 2^{2n} = (2^2)^n = 4^n > (n+1)(n+2)\ldots(n+n) > (n+1)(n+2) \ldots (2n)$$ since each term in the product is less than 4. Now compare \(N\) with \((2n+1)N\). Since \((2n+1)>1\), we have $$ (2n+1)N > N = (n+1)(n+2) \ldots (2n).$$

Conclusion and result

Combining the results from Step 2, we have $$ 2^{2n} > (n+1)(n+2) \ldots (2n) = N > (2n+1)N.$$ Thus, we have shown that $$ N=C(2 n, n)=\frac{(n+1)(n+2) \ldots(n+n)}{n !}<2^{2 n}<(2n+1)N.$$

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that studies the counting, arrangement, and combination of objects. A core element of combinatorics is the concept of binomial coefficients, which are often denoted by \( C(n, k) \) or \( \binom{n}{k} \), representing the number of ways to choose \( k \) objects from \( n \) objects without regard to order. The binomial coefficient can be calculated using the formula:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

This formula represents an important concept as it applies factorials to solve problems related to counting and combinations.

In the original problem, the task is to prove inequalities involving binomial coefficients \( C(2n, n) \). Here, combinatorics helps us understand how many ways there are to choose \( n \) elements from \( 2n \), which is fundamental in solving the inequality provided. Understanding combinatorics enables us to structure these solutions in a mathematically valid and efficient way.

Factorials

Factorials are fundamental in combinatorics, providing a way to express permutations and combinations in mathematical terms. The factorial of a non-negative integer \( n \), denoted \( n! \), is the product of all positive integers less than or equal to \( n \), i.e., \( n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1 \).

This concept is vital when dealing with binomial coefficients. In our exercise, we utilized factorials extensively to simplify and transform the expression \( C(2n, n) = \frac{(2n)!}{n!n!} \). By breaking down and rearranging the terms, we enhance our ability to prove inequalities like \( N < 2^{2n} \) and \( 2^{2n} < (2n+1)N \).

• Factorials simplify complex permutations.
• Involved in the binomial coefficient formula to make calculations manageable.

Mastering the use of factorials is essential to maneuver through many combinatorial problems efficiently.

Inequalities in Number Theory

In number theory, inequalities are used to compare sizes of numbers or expressions, providing crucial insights in proofs and solutions. They allow us to establish boundaries and relationships between different mathematical expressions. In the given exercise, we are tasked with proving that:

• \( N = \frac{(n+1)(n+2) \ldots (2n)}{n!} < 2^{2n} < (2n+1)N \)

Understanding inequalities requires us to grasp how to determine when one expression is smaller or larger than another and under what conditions these relationships hold true.

Through the properties of numbers and expressions derived from concepts like factorials and binomial coefficients, we can mathematically demonstrate that these inequalities are valid. This often involves manipulating expressions and employing various algebraic strategies to show conditions under which these inequalities maintain their truth. By mastering inequalities, we gain tools that help us explore deeper mathematical structures and proofs.