Use the formula for nCr to evaluate each expression. ₇C₇

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Recall the formula for combinations, which is given by $nCr = \frac{n!}{r!(n-r)!}$, where $n!$ denotes the factorial of $n$.

Identify the values of $n$ and $r$ from the expression $\binom{7}{7}$, so here $n = 7$ and $r = 7$.

Substitute these values into the formula: $\binom{7}{7} = \frac{7!}{7!(7-7)!}$.

Simplify the factorial in the denominator: $7 - 7 = 0$, so the expression becomes $\frac{7!}{7! \times 0!}$.

Recall that \$0!$ is defined as 1, so the expression simplifies to \(\frac{7!}{7! \times 1}$, which can be further simplified by canceling \)7!$ in numerator and denominator.

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주요 개념

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Combination Formula (nCr)

The combination formula nCr calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is fundamental for solving problems involving selections or subsets.

Factorial Function

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in the combination formula to calculate permutations and combinations.

Special Case: nCr when r = n

When the number of items chosen r equals the total number n, the combination nCr equals 1 because there is exactly one way to choose all items from the set. This simplifies calculations and helps quickly evaluate expressions like 7C7.

Use the formula for nCr to evaluate each expression. 7C7