Evaluate each expression.7P33!−7C3\frac{_7P_3}{3!} - _7C_3

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 21

9장, 문제 21

Evaluate each expression.
$\frac{_{7} P_{3}}{3!} -_{7} C_{3}$

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Identify the components of the expression: $\frac{^7P_3}{3!} - 7C_3$. Here, $^7P_3$ represents the number of permutations of 7 items taken 3 at a time, \$3!$ is the factorial of 3, and $7C_3$ is the number of combinations of 7 items taken 3 at a time.

Recall the formulas for permutations and combinations: - Permutations: $^nP_r = \frac{n!}{(n-r)!}$ - Combinations: $^nC_r = \frac{n!}{r!(n-r)!}$

Calculate $^7P_3$ using the permutation formula: $^7P_3 = \frac{7!}{(7-3)!} = \frac{7!}{4!}$.

Calculate \$3!$ which is the factorial of 3: $3! = 3 \times 2 \times 1$.

Calculate \$7C_3$ using the combination formula: $7C_3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}$.

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주요 개념

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

Permutation

A permutation refers to the arrangement of objects in a specific order. The notation nPr represents the number of ways to arrange r objects from a set of n distinct objects, calculated as n! / (n-r)!. Understanding permutations is essential when order matters in selection.

Combination

A combination is a selection of objects where order does not matter. The notation nCr represents the number of ways to choose r objects from n distinct objects, calculated as n! / [r!(n-r)!]. Combinations are used when the arrangement order is irrelevant.

Factorial and Simplification of Expressions

Factorials (n!) represent the product of all positive integers up to n and are fundamental in calculating permutations and combinations. Simplifying expressions involving factorials, permutations, and combinations requires careful manipulation to reduce complex terms and evaluate the expression correctly.

Evaluate each expression.7P33!−7C3\(\frac{_7P_3}{3!}\) - _7C_3