In Exercises 9–16, use the formula for nCr to evaluate each expression. ₉C₅

Verified step by step guidance

1. Recall the formula for combinations, which is given by: $n_r=\frac{n!}{r!\cdot (n-r)!}$, where $n$ is the total number of items and $r$ is the number of items chosen.
2. Identify the values of $n$ and $r$ from the problem: here, $n=\; 9$ and $r=\; 5$.
3. Substitute these values into the formula: $\frac{9!}{5!\cdot (9-5)!}$.
4. Simplify the factorial expressions where possible, especially the factorial in the denominator: $(9-5)!\; =\; 4!$.
5. Calculate the factorial values step-by-step or use simplification techniques to reduce the expression before computing the final value.

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.