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Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x − 1)5
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 9, Problema 26
In Exercises 23–28, evaluate each factorial expression. 20!/2!18!
Guida verificata passo dopo passo1
Identify the factorial expression: \( \frac{20!}{2!18!} \).
Recognize that \( n! \) (n factorial) is the product of all positive integers up to \( n \).
Notice that the expression can be simplified by canceling common terms in the numerator and denominator.
Rewrite the expression: \( \frac{20 \times 19 \times 18!}{2! \times 18!} \).
Cancel \( 18! \) from the numerator and denominator, then simplify the remaining expression: \( \frac{20 \times 19}{2!} \).
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Factorial Definition
A factorial, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations, combinations, and various mathematical calculations, making them fundamental in algebra.
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Factorials
Binomial Coefficient
The expression n!/(k!(n-k)!) represents the binomial coefficient, which counts the number of ways to choose k elements from a set of n elements without regard to the order of selection. This is crucial in combinatorics and is often written as C(n, k) or 'n choose k'.
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Simplifying Factorials
When evaluating expressions involving factorials, simplification is key. For instance, in the expression 20!/2!18!, the 20! can be rewritten as 20 × 19 × 18!, allowing for cancellation of the 18! in the numerator and denominator, simplifying the calculation significantly.
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5:22Factorials
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