Domanda del libro di testo
Write the first three terms in the binomial expansion, expressing the result in simplified form. (x-3)^9
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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 73
Evaluate the expression. *permutation notation* the number of permutations 9 things taken 5 at a time (sub 9)P(sub 5)
Guida verificata passo dopo passo1
Understand the permutation formula: The number of permutations of n things taken r at a time is given by the formula: , where n! is the factorial of n.
Identify the values of n and r from the problem: Here, n = 9 and r = 5. This means we are calculating the number of permutations of 9 things taken 5 at a time.
Substitute the values of n and r into the formula: .
Simplify the denominator: Calculate , which is . The formula now becomes: .
Expand and simplify the factorials: Write out as , and cancel the in the numerator and denominator. This leaves: .
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Permutations
Permutations refer to the different ways in which a set of items can be arranged or ordered. In mathematics, the number of permutations of 'n' items taken 'r' at a time is calculated using the formula P(n, r) = n! / (n - r)!, where 'n!' denotes the factorial of 'n'. This concept is crucial for understanding how to count arrangements when the order of selection matters.
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7:11
Introduction to Permutations
Factorial
A factorial, denoted as 'n!', is the product of all positive integers up to 'n'. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, particularly in calculating permutations and combinations, as they provide a way to quantify the total arrangements of a set of items.
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5:22Factorials
Combination vs. Permutation
The distinction between combinations and permutations is essential in combinatorial mathematics. While permutations consider the order of selection (e.g., ABC is different from ACB), combinations focus solely on the selection itself, disregarding order (e.g., ABC is the same as ACB). Understanding this difference is vital when determining which formula to apply in problems involving arrangements.
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3:18Permutations vs. Combinations
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