Problem 55 - College Algebra

Question

Use the formula for \(_{n} C_{r}\) to solve To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers ( 1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Step-by-Step Solution

Answer

The number of different combinations of selecting 6 numbers out of 53 numbers is \(_{53} C_{6} = 22,957,480 \)

Step 1: Understanding the Combination formula

The combination formula \(_{n}C_{r} = \frac{n!}{r!(n-r)!}\) is used when we want to find out how many ways 'r' items can be selected from 'n' items and the order of selection does not matter.

Step 2: Identify 'n' and 'r'

In the lottery problem, 'n' is the total number of numbers that can be selected which is 53. 'r' is the number of numbers to be selected to win the game, which is 6.

Step 3: Plug in values into the formula

Fill the identified values of 'n' and 'r' into the combination formula.

Step 4: Calculate factorials

Calculate the values of 53!, 6! and (53-6)!. Do not try to calculate 53! directly because it's a huge number and it's easier to cancel out terms where possible.

Step 5: Calculate the number of selections

Evaluate the expression to get the number of different selections possible.

Key Concepts

Combination Formula

The combination formula, denoted as \( _{n}C_{r} \), is a fundamental principle in combinatorics.It is used to determine how many different groups of 'r' items can be formed from a larger set of 'n' items, where the order of selection does not matter.
If you think of selecting items for a committee or forming lottery numbers, the combination formula is indispensable.The formula is given by:

• \( _{n}C_{r} = \frac{n!}{r!(n-r)!} \)

This expression helps in computing the number of ways to choose 'r' elements from a set of 'n' elements.
The significance of order not mattering differentiates combinations from permutations, where order is important.

Factorials

Factorials are mathematical expressions that give you the product of all positive integers up to a specific number.Represented by the exclamation point symbol, '!', they are a way to easily compute large numerical expressions.
For instance, 'n!' means multiply all whole numbers from 1 up to 'n'.

• 0! = 1, by definition
• 5! = 5 \( \times \) 4 \( \times \) 3 \( \times \) 2 \( \times \) 1 = 120

Factorials grow very fast as 'n' increases, which is why directly computing large factorials, like 53! in our exercise, is not feasible without simplification.
In the combination formula, these factorials work together to help count the combinations efficiently by canceling out large portions of the numbers.

Lottery Probability

Lottery probability, in this context, refers to the chances of winning a game where you need to pick a specific subset of numbers. In the Florida LOTTO example, it involves selecting 6 numbers out of 53 without regard to sequence. To calculate the probability of achieving a winning combination, we use the combination formula described earlier.
The result will tell us how many potential selections exist, but remember, this number also indicates how rare a winning combination is. As lotteries are designed to have low probabilities of winning, knowing the total combinations informs players about their odds in a clear mathematical way.

Number Selection

Number selection is a crucial aspect of solving lottery-related problems. In the exercise, the numbers are drawn from a set ranging from 1 to 53, and you need to select a subset of 6. Since the order of the selected numbers does not matter, choosing a combination over a permutation is appropriate.
The process of selecting numbers can be likened to picking cards from a deck or choosing ingredients in a recipe where the outcome is the group of items, not their order.

• Think of number selection as forming a team where each member represents a number.
• Focus on which numbers are chosen rather than the sequence they are picked in.

This approach simplifies the calculation and aids in using the combination formula accurately.