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 90

Chapter 9, Problem 90

Evaluate n!/(n-r)!r! for n = 8 and r = 3

Verified step by step guidance

Recognize that the expression given, \( \frac{n!}{(n-r)!r!} \), is the formula for combinations, often written as \( \binom{n}{r} \), which calculates the number of ways to choose \( r \) objects from \( n \) without regard to order.

Substitute the given values \( n = 8 \) and \( r = 3 \) into the formula to get \( \frac{8!}{(8-3)! \times 3!} \).

Simplify the factorial in the denominator: \( (8-3)! = 5! \), so the expression becomes \( \frac{8!}{5! \times 3!} \).

Write out the factorials explicitly or use the property that \( \frac{8!}{5!} = 8 \times 7 \times 6 \) to simplify the numerator and denominator before multiplying by \( 3! \).

Calculate \( 3! = 3 \times 2 \times 1 \), then divide the product from the numerator by this value to find the final simplified result.

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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 fundamental in permutations and combinations, representing the number of ways to arrange or select items.

Combinations Formula

The expression n! / [(n - r)! r!] calculates combinations, representing the number of ways to choose r items from n without regard to order. It is read as 'n choose r' and is essential in probability and counting problems.

Substitution and Simplification

To evaluate the combination formula for specific values, substitute n and r into the formula and simplify step-by-step. This involves calculating factorials and reducing the expression to find the numerical result.

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Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1)to find the seventh term of the sequence 11, 33, 99, 297,...

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