Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n2
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 14
Use the formula for nCr to evaluate each expression. 4C4
Verified step by step guidance1
Recall the formula for combinations, which is used to find the number of ways to choose r objects from a set of n objects without regard to order: \[ \displaystyle {n \choose r} = \frac{n!}{r! (n-r)!} \]
Identify the values of n and r from the problem: here, \[ n = 4 \] and \[ r = 4 \].
Substitute the values of n and r into the combination formula: \[ \displaystyle {4 \choose 4} = \frac{4!}{4! (4-4)!} \].
Simplify the factorial expressions in the denominator: \[ (4-4)! = 0! \], and recall that by definition, \[ 0! = 1 \].
Write the simplified expression ready for evaluation: \[ \displaystyle {4 \choose 4} = \frac{4!}{4! \times 1} \].
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combination Formula (nCr)
The combination formula calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is fundamental for solving problems involving selections or groups.
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5:22
Combinations
Factorial Function
A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials are used in permutations and combinations to count arrangements and selections systematically.
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5:22Factorials
Evaluating Special Cases in Combinations
When r equals n in nCr, the combination equals 1 because there is exactly one way to choose all items from the set. Understanding these special cases helps simplify calculations and avoid unnecessary computation.
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5:22Combinations
Related Practice
Textbook Question
In Exercises 12–15, write the first six terms of each arithmetic sequence. a1 = 3/2, d = -1/2
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Textbook Question
Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a6 when a1 = 13, d = 4.
Textbook Question
Use the formula for nCr to evaluate each expression. 5C0
Textbook Question
Write the first six terms of each arithmetic sequence. an = an-1 -0.4, a1 = 1.6
Textbook Question
The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a1=7 and an=an-1 + 5 for n≥2
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