Problem 8 - Beginning and Intermediate Algebra

Question

Fill in each blank with the correct response. For any nonnegative integer \(n,\) the binomial coefficient \({ }_{n} C_{n}\) is equal to ______.

Step-by-Step Solution

Answer

1

Step 1: Understanding Binomial Coefficient

The binomial coefficient \({ }_{n} C_{n}\) represents the number of ways to choose \( n \) elements from \( n \) elements. It is also read as 'n choose n'.

Step 2: Using the Binomial Coefficient Formula

The general formula for the binomial coefficient is given by: \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \). For \({ }_{n} C_{n}\), substitute \( k = n \).

Step 3: Simplifying the Expression

Substitute \( k = n \) in the formula: \({ }_{n} C_{n} = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \times 0!}\).

Step 4: Evaluating Factorials

Recall that \( 0! = 1 \). Therefore: \(\frac{n!}{n! \times 1} = \frac{n!}{n!} = 1 \).

Key Concepts

factorials

A factorial is a function represented by an exclamation mark (!) after a number. It multiplies that number by all the smaller natural numbers down to 1. This means that for a positive integer , the factorial of is calculated as follows: