Domanda del libro di testo
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 a12 when a1 = 5, r = - 2
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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 11
Use the formula for nCr to evaluate each expression. 11C4
Guida verificata passo dopo passo1
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: \[{{n}\choose{r}} = \frac{n!}{r!(n-r)!}\]
Identify the values of n and r from the problem: here, \(n = 11\) and \(r = 4\).
Substitute these values into the formula: \[{{11}\choose{4}} = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!}\]
Simplify the factorial expressions by expanding only the necessary parts of the factorials to make calculation easier: \[\frac{11 \times 10 \times 9 \times 8 \times 7!}{4! \times 7!}\]
Cancel the common \$7!$ terms in numerator and denominator, then simplify the remaining expression by calculating the numerator and denominator separately before dividing.
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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 essential for solving problems involving selections or groups.
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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, 4! = 4 × 3 × 2 × 1 = 24. Factorials are used in the combination formula to calculate permutations and combinations, making them fundamental in counting problems.
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5:22Factorials
Evaluating Combinations
To evaluate a combination like 11C4, substitute n = 11 and r = 4 into the formula and compute the factorial values. Simplifying factorial expressions by canceling common terms can make calculations easier. This process yields the total number of unique groups of 4 from 11 items.
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5:22Combinations
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