Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 85

Chapter 9, Problem 85

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Factorials

Related Practice

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Solve: $6 ∣ 1 – 2 x ∣ — 7 = 11$ .

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Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

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Write an equation in point-slope form and slope-intercept form for the line passing through (-2, -6) and perpendicular to the line whose equation is x − 3y+ 9 = 0.

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Show that the sum of the first n positive odd integers,1 +3 +5 + ··· + (2n − 1), ... is n².

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Textbook Question

Use a calculator's factorial key to evaluate each expression. 20!/(20−3)!

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Textbook Question

Solve: log₂ (x+9) — log₂ x = 1.

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

Verified video answer for a similar problem:

Key Concepts

Factorials

Permutation Formula

Using a Calculator for Factorials

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