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Chapter 11: Problem 51

A restaurant lunch special allows the customer to choose two vegetables from this list: okra, corn, peas, carrots, and squash. How many outcomes are possible if the customer chooses two different vegetables?

Short Answer

Expert verified
The number of possible outcomes or combinations of choosing 2 different vegetables from a list of 5 is 10.

Step by step solution

01

Identifying the problem

This problem is a combination problem. Combinations are used when the order of the selection does not matter. Here, we want to find the number of combinations of 2 vegetables that can be chosen from a list of 5 vegetables.
02

Using the combination formula

The formula for combinations is \( C(n,k) = \frac{n!}{k!(n-k)!} \) where \( n! \) is the factorial of \( n \), and \( k \) and \( n \) are the total number of elements and the number of elements to select, respectively. The symbol \( ! \) indicates a factorial, meaning multiplying all positive integers up to that number.
03

Applying the combination formula

We apply the formula to find the number of combinations of 2 vegetables from 5. So, \( n=5 \) and \( k=2 \), therefore, the number of combinations = \( \frac{5!}{2!(5-2)!} \).
04

Calculating the Factorials

Firstly, calculate the factorial of 5, 2! and (5-2)!. So, 5! = 5 * 4 * 3 * 2 * 1 = 120, 2! = 2 * 1 = 2 and 3! = 3 * 2 * 1 = 6.
05

Substituting the factorials

Substitute the factorial values into the combination formula: \( \frac{120}{2*6} \). Simplify the equation to find the result.
06

Final Calculation

Calculate the division to obtain the number of combinations: 120 / 12 = 10. This means there are 10 different ways to choose 2 vegetables from the 5.

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