Use the formula for nPr to evaluate each expression. ₆P₆

검증된 단계별 안내

Recall the formula for permutations: $nP_r = \frac{n!}{(n-r)!}$, where $n$ is the total number of items and $r$ is the number of items to arrange.

Identify the values of $n$ and $r$ in the expression \$6P6$. Here, $n = 6$ and $r = 6$.

Substitute these values into the formula: $6P6 = \frac{6!}{(6-6)!}$.

Simplify the denominator: $(6-6)! = 0!$. Remember that by definition, $0! = 1$.

The expression now becomes $6P6 = \frac{6!}{1} = 6!$. To find the value, you would calculate $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$, but the problem only asks for the setup.

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주요 개념

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

Permutation Formula (nPr)

The permutation formula nPr calculates the number of ways to arrange r objects from a set of n distinct objects, where order matters. It is given by nPr = n! / (n - r)!, where '!' denotes factorial.

Factorial Notation

Factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Factorials are essential in permutations and combinations calculations.

Evaluating nPr When r = n

When the number of objects chosen r equals the total number n, nPr simplifies to n!, because (n - n)! = 0! = 1. Thus, nPn = n!, representing all possible arrangements of the entire set.

Use the formula for nPr to evaluate each expression. 6P6