교과서 질문
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.
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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
9장, 문제 10
Use the formula for nCr to evaluate each expression. 10C6
검증된 단계별 안내1
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 \(\binom{10}{6}\).
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
5:22Factorials
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