In Exercises 45-48, use combinations and permutations.
48. An employer must hire 2 people from a list of 13 applicants. In how many ways can the employer choose to hire the two people?

Verified step by step guidance

1. Step 1: Recognize that the problem involves selecting 2 people from a group of 13 without regard to the order in which they are chosen. This indicates that the problem involves combinations, not permutations.
2. Step 2: Recall the formula for combinations, which is given by: $C(n,r)=\frac{n!}{r!(n-r)!}$, where $n$ is the total number of items (applicants in this case), and $r$ is the number of items to choose (2 people in this case).
3. Step 3: Substitute the values $n=13$ and $r=2$ into the formula: $C(13,2)=\frac{13!}{2!(13-2)!}$.
4. Step 4: Simplify the factorials in the formula. Start by calculating $13!$, $2!$, and $11!$. Then cancel out the common terms in the numerator and denominator.
5. Step 5: Perform the division to find the total number of ways the employer can choose 2 people from 13 applicants. The result will be the value of $C(13,2)$.

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where the order of selection does not matter. In this context, when hiring 2 people from 13 applicants, we are interested in how many unique groups of 2 can be formed, regardless of the order in which they are chosen.

Factorial

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers up to n. Factorials are essential in calculating combinations and permutations, as they help determine the total arrangements of items. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120.

Binomial Coefficient

The binomial coefficient, often represented as C(n, k) or n choose k, quantifies the number of ways to choose k items from n items without regard to the order of selection. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), which is crucial for solving the hiring problem by determining how many ways 2 applicants can be selected from 13.