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

검증된 단계별 안내

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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주요 개념

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

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.