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 10
Use the formula for nCr to evaluate each expression. 10C6
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:
\[ \text{nCr} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Identify the values of \(n\) and \(r\) from the problem. Here, \(n = 10\) and \(r = 6\).
Substitute these values into the formula:
\[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} \]
Simplify the factorial expressions by expanding only as much as needed to cancel terms. For example, write \$10!\( as \(10 \times 9 \times 8 \times 7 \times 6!\) so that \)6!$ cancels out in numerator and denominator.
After canceling, you will have a fraction with multiplication in numerator and denominator. Multiply the numbers in numerator and denominator separately, then divide to find the value of \(\binom{10}{6}\).
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.
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5:22
Combinations
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 computing combinations and permutations.
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5:22Factorials
Simplifying Factorial Expressions
When evaluating combinations, simplifying factorial expressions by canceling common terms in numerator and denominator helps reduce calculation complexity. For example, 10! / 6! can be simplified by expanding only necessary terms.
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5:22Factorials
Related Practice
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
Use the formula for nCr to evaluate each expression. 11C4
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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
In Exercises 11–16, a die is rolled. Find the probability of getting a 4.
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Textbook Question
In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=(−1)n+1/(2n−1)