   Chapter 2.2, Problem 40E

Chapter
Section
Textbook Problem

Exercise 37 − 39 can be generalized as follows: If 0 ≤ k ≤ n and the set A has n elements, then the number of elements of the power set P ( A ) containing exactly k elements is ( n k ) .Use this result to write an expression for the total number of elements in the power set P ( A ) .Use the binomial theorem as stated in Exercise 25 to evaluate the expression in part a and compare this result to exercise 27 and 37. (Hint: set a = b = 1 in the binomial theorem.)If n is a nonnegative integer and the set A has n elements, then the power set P ( A ) has 2 n If n ≥ 2 and the set A has n elements, then the number of elements of the power set P ( A ) containing exactly two elements is ( n 2 ) = n ( n − 1 ) 2 .If n ≥ 3 and the set A has n elements, then the number of elements of the power set P ( A ) containing exactly three elements is ( n 3 ) = n ( n − 1 ) ( n − 2 ) 3 ! .Let a and b be a real number, and let n be integers with 0 ≤ r ≤ n . The binomial theorem states that ( a + b ) n = ( n 0 ) a n + ( n 1 ) a n − 1 b + ( n 2 ) a n − 2 b 2 + ... + ( n r ) a n − r b r + ....... + (       n n − 2 ) a 2 b n − 2 + ( n n − 1 ) a b n − 1 + ( n n ) b n = ∑ r = 0 n ( n r )     a n − r b r Where the binomial coefficients ( n r ) are defined by ( n r ) = n ! ( n − r ) ! r ! ,With r ! = r ( r − 1 ) ......... ( 2 ) ( 1 ) for r ≥ 1 and 0 ! = 1 . Prove that the binomial coefficients satisfy the equation (     n r − 1 ) + ( n r ) = ( n + 1       r ) for 1 ≤ r ≤ n

(a)

To determine

An expression for the total number of elements in the power set P(A).

Explanation

Given information:

Result: If 0kn and the set A has n elements, then the number of elements of the power set P(A) containing exactly k elements is (nk).

Explanation:

It is given that if 0kn and the set A has n elements, then the number of elements of the power set P(A) containing exactly k elements is (nk).

The number of elements of the power set P(A) containing exactly 0 elements is (n0)

The number of elements of the power set P(A) containing exactly 1 elements is (n1

(b)

To determine

The value of expression in part a by using binomial theorem.

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find the constants m and b in the linear function f(x) = mx + b such that f(0) = 2 and f(3) = 1.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

For the following scores, find the value of each expression: X 3 2 4 2 a. X b. (X)2 c. X 2 d. (X 2)

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Finding a Limit In Exercises 5-18, find the limit. limx13x+5x+1

Calculus: Early Transcendental Functions (MindTap Course List)

Find f(x) if f(x) = 10x2 + cos x. a) 20x sin x + C b) 20x cos x + C c) 103x3cosx+C d) 103x3+sinx+C

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 