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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 8 - Sequences, Induction, and Probability문제 55
9장, 문제 55

Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

검증된 단계별 안내
1
Identify the problem as a combination problem because the order of selection does not matter. We need to find the number of ways to choose 6 numbers from 53 without regard to order.
Recall the formula for combinations (nCr), which is given by: \[ \text{nCr} = \frac{n!}{r!(n-r)!} \] where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose.
Substitute the given values into the formula: Here, \(n = 53\) and \(r = 6\), so the expression becomes \[ \text{53C6} = \frac{53!}{6!(53-6)!} = \frac{53!}{6! \times 47!} \]
Understand that the factorial notation (\(n!\)) means the product of all positive integers from 1 up to \(n\). For example, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
To simplify the calculation, expand the numerator \$53!$ only down to $(53-6+1) = 48$, so it becomes \(53 \times 52 \times 51 \times 50 \times 49 \times 48!\), which allows the \$48!$ in numerator and denominator to cancel out, leaving: \[ \frac{53 \times 52 \times 51 \times 50 \times 49 \times 48!}{6! \times 47!} = \frac{53 \times 52 \times 51 \times 50 \times 49 \times 48!}{6! \times 47!} \] Then cancel \$48!$ with part of \$47!$ accordingly to simplify further before calculating.

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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 items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is essential for problems where order does not matter, such as selecting lottery numbers.
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 used in the combination formula to count permutations and combinations by accounting for the number of ways to arrange or select items.
추천 영상:
5:22
Factorials

Order Irrelevance in Combinations

In combinations, the order of selection does not matter, meaning that selecting numbers {1, 2, 3} is the same as {3, 2, 1}. This contrasts with permutations, where order matters. Understanding this distinction helps determine when to use combinations versus permutations in counting problems.
추천 영상:
5:22
Combinations
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