Short Answer

Show that $(\begin{array}{l} n \\ k \end{array}) \leq 2^{n}$ for all positive integers n and all integers k with $0 \leq k \leq n$ .

For all positive integers n and all integers k it's proven that $(\begin{array}{l} n \\ k \end{array}) \leq 2^{n}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use Binomial theorem

Binomial theorem: binomial theorem, statement that for any positive integer n , the nth power of the sum of two numbers x and y may be expressed as the sum of n + 1 terms of the form.

$(x + y)^{n} = \sum_{j = 0}^{n} (\begin{array}{l} n \\ j \end{array}) x^{n - j} y^{j}$

To proof:

$(\begin{array}{l} n \\ k \end{array}) \leq 2^{n}$

Step 2: For all positive integers and all integers with 0≤k≤n

Let n be a positive integer and an k integer with $0 \leq k \leq n$

$\begin{aligned} 2^{n} = (1 + 1)^{n} \\ = \sum_{j = 0}^{n} (\begin{array}{c} n \\ j \end{array}) (1)^{n - j} (1)^{j} \\ = \sum_{j = 0}^{n} (\begin{array}{c} n \\ j \end{array}) (1) (1) \\ = \sum_{j = 0}^{n} (\begin{array}{c} n \\ j \end{array}) \\ \geq (\begin{matrix} n \\ k \end{matrix}) \end{aligned}$

Thus, . $(\begin{array}{l} n \\ k \end{array}) \leq 2^{n}$

Find Study Materials for

Create Study Materials

Select your language

Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Show that $(\begin{array}{l} n \\ k \end{array}) \leq 2^{n}$ for all positive integers n and all integers k with $0 \leq k \leq n$ .

Step by Step Solution

Step 1: Use Binomial theorem

Step 2: For all positive integers and all integers with 0≤k≤n

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Discrete Mathematics and its Applications

Answers without the blur.

Short Answer

Show that (nk)≤2n for all positive integers n and all integers k with 0≤k≤n .

Step by Step Solution

Step 1: Use Binomial theorem

Step 2: For all positive integers and all integers with 0≤k≤n

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Show that $(\begin{array}{l} n \\ k \end{array}) \leq 2^{n}$ for all positive integers n and all integers k with $0 \leq k \leq n$ .