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

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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: $nCr = \frac{n!}{r!(n-r)!}$.

Identify the values of n and r from the problem: here, $n = 9$ and $r = 5$.

Substitute these values into the formula: $9C5 = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!}$.

Write out the factorial expressions explicitly or simplify by canceling common terms in the numerator and denominator to make calculations easier.

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).

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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, 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.

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