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

Verified step by step guidance

1. Recall the formula for combinations, which is given by \(nCr = \frac{n!}{r!(n-r)!}\), where \(n!\) denotes the factorial of \(n\).
2. Identify the values of \(n\) and \(r\) from the expression \(\binom{7}{7}\), so here \(n = 7\) and \(r = 7\).
3. Substitute these values into the formula: \(\binom{7}{7} = \frac{7!}{7!(7-7)!}\).
4. Simplify the factorial in the denominator: \$7 - 7 = 0\(, so the expression becomes \)\frac{7!}{7! \times 0!}$.
5. 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.

Key Concepts

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.