Use the formula for nCr to evaluate each expression. ₁₁C₄

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Recall the formula for combinations, which is used to find the number of ways to choose r objects from a set of n objects without regard to order: \[{{n}\choose{r}} = \frac{n!}{r!(n-r)!}\]

Identify the values of n and r from the problem: here, $n = 11$ and $r = 4$.

Substitute these values into the formula: \[{{11}\choose{4}} = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!}\]

Simplify the factorial expressions by expanding only the necessary parts of the factorials to make calculation easier: \[\frac{11 \times 10 \times 9 \times 8 \times 7!}{4! \times 7!}\]

Cancel the common \$7!$ terms in numerator and denominator, then simplify the remaining expression by calculating the numerator and denominator separately before dividing.

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주요 개념

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Combination Formula (nCr)

The combination formula, denoted as 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 essential for solving problems involving selections or groups.

Factorials

A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials are used in the combination formula to calculate permutations and combinations, making them fundamental in counting problems.

Evaluating Combinations

To evaluate a combination like 11C4, substitute n = 11 and r = 4 into the formula and compute the factorial values. Simplifying factorial expressions by canceling common terms can make calculations easier. This process yields the total number of unique groups of 4 from 11 items.

Use the formula for nCr to evaluate each expression. 11C4