Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 85

9장, 문제 85

In Exercises 81–85, use a calculator's factorial key to evaluate each expression.
$\frac{54!}{(54 - 3)! 3!}$

검증된 단계별 안내

Recognize that the expression

\frac{54^{!}}{{(54-3)}^{!} 3!}

is a combination formula in disguise, specifically

C(n, r) = \(\frac{n!}{(n-r)!r!}\)

where

n=54

and

r=3

Rewrite the expression as

C(54, 3) = \(\frac{54!}{(54-3)!3!}\) = \(\frac{54!}{51!3!}\)

to clearly see the combination structure.

Understand that factorial notation means the product of all positive integers up to that number, for example,

54! = 54 \(\times\) 53 \(\times\) 52 \(\times\) 51!.

This allows simplification by canceling

51!

in numerator and denominator.

Simplify the fraction by canceling

51!

from numerator and denominator, leaving

\(\frac{54 \times 53 \times 52}{3!}\)

Calculate

3! = 3 \(\times\) 2 \(\times\) 1 = 6

, then divide the product

54 \(\times\) 53 \(\times\) 52

by 6 to find the value of the expression.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Factorials

A factorial, denoted 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 commonly used in permutations, combinations, and probability calculations.

Permutation Formula

Permutations count the number of ways to arrange r objects from a set of n distinct objects, given by n! / (n - r)!. This formula accounts for order and is essential for problems involving arrangements or sequences.

Using a Calculator for Factorials

Many scientific calculators have a factorial function (n!) that allows quick computation of large factorials. This is useful for evaluating expressions like 54! / (54−3)! 3! without manual multiplication.

In Exercises 81–85, use a calculator's factorial key to evaluate each expression.54!(54−3)!3!\(\frac{54!}{(54-3)!3!}\)