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When dealing with problems of counting ways to choose elements, it is important to know whether the order of selection matters. Permutations and combinations are two different methods for counting these ways.
Permutations consider the order of the elements. For example, choosing two numbers from {1, 2, 3} in different orders (12 vs 21) counts as separate outcomes.
Combinations, however, do not consider the order. Choosing two numbers from {1, 2, 3} would only count once, no matter the order (12 is the same as 21).
In the context of the New Jersey Pick 6 lottery game, the order of the chosen numbers does not matter. Therefore, the combination method is used to calculate the probability.

Calculating the probability of winning a lottery is a common problem in combinatorial probability. The New Jersey Pick 6 lottery involves selecting 6 different numbers from a pool of 49. For a player to win the top prize, their chosen numbers must all match the drawn numbers, with the order being irrelevant.

Here's a step-by-step approach:

• Identify the total number of possible outcomes. This can be done using the combination formula since the order does not matter.
• Calculate the number of ways to win — this is always 1 because there's only one correct set of winning numbers.
• Use the combination formula \(C(n, k)\) to find the number of possible combinations, where \(n\) is the total number of options (49) and \(k\) is the number of selections (6).
• The probability of winning is then the ratio of winning outcomes to total outcomes.

To determine the number of ways to choose a subset of items from a larger set, without considering the order, we use the combination formula:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
Here, \(n\) represents the total number of items, and \(k\) is the number of items to be chosen.
In the case of the New Jersey Pick 6 lottery, \(n \) is 49 (total possible numbers) and \(k \) is 6 (numbers picked).
You can substitute these values into the formula:
\[ C(49, 6) = \frac{49!}{6! \times 43!} \]
Factorial (!), is the product of all positive integers up to that number (e.g., \(5! = 5 \times 4 \times 3 \times 2 \times 1\)).
Using calculations or a scientific calculator, you will find the total number of ways, which yields the number of possible combinations.
This formula is especially handy for problems where the order of selection does not matter, like the lottery, thereby making the concept of combinations essential in probabilistic calculations.