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

Verified step by step guidance

1. 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: \[{{n}\choose{r}} = \frac{n!}{r!(n-r)!}\]
2. Identify the values of n and r from the problem: here, \(n = 11\) and \(r = 4\).
3. Substitute these values into the formula: \[{{11}\choose{4}} = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!}\]
4. Simplify the factorial expressions by expanding only the necessary parts of the factorials to make calculation easier: \[\frac{11 \times 10 \times 9 \times 8 \times 7!}{4! \times 7!}\]
5. Cancel the common \$7!$ terms in numerator and denominator, then simplify the remaining expression by calculating the numerator and denominator separately before dividing.

Key Concepts

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.