Evaluate each expression.4C2⋅6C118C3\frac{_4C_2 \cdot _6C_1}{_{18}$$C_3$$}

Ch. 8 - Sequences, Induction, and Probability

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

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 27

9장, 문제 27

Evaluate each expression.
$\frac{_{4} C_{2} \cdot_{6} C_{1}}{_{18} C_{3}}$

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Identify the components of the expression: $\frac{4C2 \cdot 6C1}{18C3}$. Here, $nCr$ represents the combination formula, which calculates the number of ways to choose $r$ items from $n$ items without regard to order.

Recall the combination formula: $nCr = \frac{n!}{r! (n-r)!}$, where $n!$ denotes the factorial of $n$.

Calculate each combination separately using the formula: - Calculate $4C2 = \frac{4!}{2! (4-2)!}$ - Calculate $6C1 = \frac{6!}{1! (6-1)!}$ - Calculate $18C3 = \frac{18!}{3! (18-3)!}$

Substitute the calculated values of \$4C2$, $6C1$, and $18C3$ back into the original expression $\frac{4C2 \cdot 6C1}{18C3}$.

Simplify the resulting fraction by multiplying the numerator values and then dividing by the denominator value to get the final simplified expression.

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주요 개념

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Combination Formula

The combination formula, denoted as nCr, calculates the number of ways to choose r items from n without regard to order. It is given by nCr = n! / [r!(n-r)!], where '!' denotes factorial. This concept is essential for evaluating expressions involving combinations.

Factorials

Factorials represent the product of all positive integers up to a given number n, written as n!. They are fundamental in calculating permutations and combinations, as they help determine the total number of arrangements or selections.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing expressions by factoring and canceling common terms. This skill is crucial when evaluating complex expressions like the given combination formula multiplied and divided by other terms.

Evaluate each expression.4C2⋅6C118C3\(\frac{_4C_2 \cdot _6C_1}{_{18}\)C_3}