Use the formula for nCr to evaluate each expression. ₁₀C₆

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: \[ \text{nCr} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
2. Identify the values of \(n\) and \(r\) from the problem. Here, \(n = 10\) and \(r = 6\).
3. Substitute these values into the formula: \[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} \]
4. Simplify the factorial expressions by expanding only as much as needed to cancel terms. For example, write \$10!\( as \)10 \times 9 \times 8 \times 7 \times 6!\( so that \)6!$ cancels out in numerator and denominator.
5. After canceling, you will have a fraction with multiplication in numerator and denominator. Multiply the numbers in numerator and denominator separately, then divide to find the value of \(\binom{10}{6}\).

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.

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 essential in computing combinations and permutations.

Simplifying Factorial Expressions

When evaluating combinations, simplifying factorial expressions by canceling common terms in numerator and denominator helps reduce calculation complexity. For example, 10! / 6! can be simplified by expanding only necessary terms.