Use the formula for _nC_r to evaluate each expression. ₄C₄

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 a set of n objects without regard to order: \[ \displaystyle {n \choose r} = \frac{n!}{r! (n-r)!} \]
2. Identify the values of n and r from the problem: here, \[ n = 4 \] and \[ r = 4 \].
3. Substitute the values of n and r into the combination formula: \[ \displaystyle {4 \choose 4} = \frac{4!}{4! (4-4)!} \].
4. Simplify the factorial expressions in the denominator: \[ (4-4)! = 0! \], and recall that by definition, \[ 0! = 1 \].
5. Write the simplified expression ready for evaluation: \[ \displaystyle {4 \choose 4} = \frac{4!}{4! \times 1} \].

Key Concepts

Combination Formula (nCr)

The combination formula 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 groups.

Factorial Function

A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials are used in permutations and combinations to count arrangements and selections systematically.

Evaluating Special Cases in Combinations

When r equals n in nCr, the combination equals 1 because there is exactly one way to choose all items from the set. Understanding these special cases helps simplify calculations and avoid unnecessary computation.