Problem 11 - Algebra for College Students

Question

Solve each problem. See Examples 1 and 2 . In the Florida Lottery you can win a lot of money for merely selecting 6 different numbers from the numbers 1 through 49. How many different ways are there to select the 6 numbers?

Step-by-Step Solution

Answer

13983816 ways

Step 1: Understand Combination Formula

To determine the number of ways to choose 6 different numbers from 49, use the combination formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, and \( k \) is the number of items to choose.

Step 2: Identify Values

In this problem, \( n = 49 \) and \( k = 6 \). Substitute these values into the formula.

Step 3: Substitute Values

Substitute the values of \( n = 49 \) and \( k = 6 \) into the combination formula: \[ C(49, 6) = \frac{49!}{6!(49-6)!} \]

Step 4: Simplify the Expression

Simplify the expression step by step: \[ C(49, 6) = \frac{49!}{6! \cdot 43!} \] Further simplify by canceling out the \( 43! \) from the numerator and denominator: \[ C(49, 6) = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

Step 5: Calculate the Result

Now, calculate the above expression step by step: \[ 49 \times 48 \times 47 \times 46 \times 45 \times 44 = 10068347520 \] and \[ 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \] Therefore, \[ C(49, 6) = \frac{10068347520}{720} = 13983816 \]

Key Concepts

Combination Formula

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. One of the key aspects in this field is understanding combinations. Combinations refer to selections of items where order does not matter. To calculate the number of combinations of a set of objects, we use the combination formula:
The combination formula is written as: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Here, \( n \) represents the total number of items, and \( k \) is the number of items to choose. The exclamation mark (!) represents a factorial, which we'll explain next.
When applied to problems like selecting lottery numbers, this formula helps determine how many different ways you can choose a subset of numbers from a larger set. It’s essential for situations where the order of selection does not matter, like picking 6 numbers out of 49 in a lottery.

Factorial

A factorial, denoted as \( n! \), is a product of all positive integers up to a certain number \( n \). For example, \( 5! \) (read as 'five factorial') means \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials form the backbone of the combination formula. Here's a concise breakdown:

• \( 0! = 1 \)
• \( 1! = 1 \)

Further examples include:

• \( 2! = 2\times 1 = 2 \)
• \( 3! = 3\times 2\times 1 = 6 \)

Higher numbers follow the same pattern. This mathematical operation ensures the combination formula correctly calculates the number of ways to select items from a set without considering the order. It's a powerful tool not just in lotteries, but also in statistical calculations and various fields of science.

Lottery Probability

Lottery probability is a fascinating application of combinatorial mathematics. When applying the combination formula to the lottery, we determine how many different ways we can select a set of numbers from a larger set.
For instance, in the Florida Lottery, to find the number of possible ways to select 6 numbers from 49, we use the combination formula with \( n = 49 \) and \( k = 6 \).
The calculation is: \[ C(49, 6) = \frac{49!}{6!(49-6)!} \] This simplifies to: \[ C(49, 6) = \frac{49!}{6!\times 43!} \] Which further simplifies as: \[ C(49, 6) = \frac{49 \times 48 \times 47 \times 46 \times 45 \times 44}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \] Finally, this results in 13,983,816 possible combinations.

Understanding these calculations can help you appreciate the odds involved in lottery games. It demonstrates just how challenging it is to match all 6 numbers, and underscores why winning the lottery is such a rare event!