Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)³
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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 9
Use the formula for nCr to evaluate each expression. 9C5
Verified step by step guidance1
Recall the formula for combinations, which is used to find the number of ways to choose r objects from n objects without regard to order: \(nCr = \frac{n!}{r!(n-r)!}\).
Identify the values of n and r from the problem: here, \(n = 9\) and \(r = 5\).
Substitute these values into the formula: \(9C5 = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!}\).
Write out the factorial expressions explicitly or simplify by canceling common terms in the numerator and denominator to make calculations easier.
Calculate the simplified expression step-by-step to find the value of \$9C5$ (do not compute the final number here, just set up the expression).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combination Formula (nCr)
The combination formula, denoted as nCr, 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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Factorials
A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in the combination formula to calculate permutations and combinations accurately.
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Evaluating Combinations
To evaluate a combination like 9C5, substitute n = 9 and r = 5 into the formula and simplify using factorial values. Understanding how to simplify factorial expressions by canceling common terms helps in efficiently computing the result without calculating large numbers fully.
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Related Practice
Textbook Question
In Exercises 5–10, a statement Sn about the positive integers is given. Write statements Sk and Sk+1 simplifying statement Sk+1 completely. Sn: 2 is a factor of n2 - n + 2.
Textbook Question
Write the first four terms of each sequence whose general term is given. an=2n/(n+4)
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Textbook Question
In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17
Textbook Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a8 when a1 = 6, r = 2
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Textbook Question
Write the first six terms of each arithmetic sequence. an = an-1 +6, a1 = −9
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