Problem 48 - College Algebra

Question

Use the formula for \(_{n} C\), to solve Exercises \(41-48\). To win in the New York State lottery, one must correctly select 6 numbers from 59 numbers. The order in which the selection is made does not matter. How many different selections are possible?

Step-by-Step Solution

Answer

The total different selections possible in the New York State lottery are \(44, 287,143,020\).

Step 1: Identify Parameters

Identify and assign appropriate values to the variables in the combination formula. Here, \(n = 59\) as there are 59 numbers to choose from and \(r = 6\) as we need to select 6 numbers.

Step 2: Apply the Combination Formula

Substitute the identified values into the combination formula: \[_{59} C_6 = \frac{59!}{6!(59-6)!}\].

Step 3: Calculation

First, calculate the factorial values of 59, 6, and 53 (which is 59 - 6). Now perform the division as indicated in the formula to get the final solution.

Key Concepts

Combination Formula

In combinatorics, the combination formula is used to determine the number of ways to choose a subset from a larger set when the order of selection does not matter. This is different from permutations, where order does matter. The formula is expressed as:

• \[ _{n}C_r = \frac{n!}{r!(n-r)!} \]

Here, \( n \) stands for the total number of items to choose from, and \( r \) represents the number of items to be chosen. The placement of \( n \) and \( r \) shows that the combination does not consider different arrangements of the same items.
This makes the combination formula particularly useful in situations like lotteries or choosing teams from a group.

Factorials

Factorials are a core component of the combination formula, helping us calculate the total number of ways items can be arranged. The factorial of a number \( n \) is denoted as \( n! \) and is calculated by multiplying all positive integers up to \( n \). For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials grow very quickly, which is why they are critical in combinatorics for calculating large sets and permutations.
When dealing with combinations, factorials allow us to "trim down" the total arrangements (permutations) to just those that fit the selection criteria, by removing the excess option that order introduces.

Lottery Odds

Lottery odds highlight the probability of winning a lottery, and they are essentially a question of calculating combinations. For instance, in the New York State lottery, you choose 6 numbers out of 59. Using combinations, you calculate how many different sets of 6 numbers can emerge from the 59. This is done using:

• \( _{59}C_6 = \frac{59!}{6!(53)!} \)

These odds show you how incredibly massive the pool of possible choices is. Hence, why winning a lottery is considered a game of chance. Lottery odds assist players in understanding just how rare a winning combination truly is, making strategic choices nearly impossible due to the vast possible outcomes.

Binomial Coefficients

Binomial coefficients are another term for combinations. These coefficients appear in the binomial theorem, which describes the algebraic expansion of powers of a binomial. The standard notation used for these coefficients is \( _{n}C_r \).

• For example, in the expression: \( (x + y)^n \), binomial coefficients give the coefficients of the terms that arise from expanding the expression.

In lotteries and other selection processes, binomial coefficients define the number of possible ways to draw certain items from a set, providing practical application beyond just theoretical math.
Understanding this helps in contexts where you need to break down problems into manageable sets of options, thus using these coefficients efficiently in both combinatorics and probability calculations.