Use the formula for nCr to evaluate each expression. ₉C₅

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 n objects without regard to order: \(nCr = \frac{n!}{r!(n-r)!}\).
2. Identify the values of n and r from the problem: here, \(n = 9\) and \(r = 5\).
3. Substitute these values into the formula: \(9C5 = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!}\).
4. Write out the factorial expressions explicitly or simplify by canceling common terms in the numerator and denominator to make calculations easier.
5. Calculate the simplified expression step-by-step to find the value of \$9C5$ (do not compute the final number here, just set up the expression).

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, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in the combination formula to calculate permutations and combinations by counting arrangements.

Evaluating Combinations

To evaluate a combination like 9C5, substitute n = 9 and r = 5 into the formula and simplify using factorial values. Understanding how to simplify factorial expressions and cancel common terms helps efficiently compute the result without calculating large numbers fully.