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Understanding combinations is crucial in calculating probabilities where the order of selection does not matter. A combination is a way to select items from a larger pool, without caring about the order in which they are picked. It can be calculated using the formula: \( C(n, k) = \frac{n!}{k!(n-k)!} \) Where: * \( n \) = total number of items * \( k \) = number of items to pick Here's how it works: * Factorial (\( ! \)) represents the product of all positive integers up to that number. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. In our exercise, we used this to determine the number of ways to choose 5 out of 75 numbers. By substituting \( n = 75 \) and \( k = 5 \), we calculated: \( C(75, 5) = \frac{75!}{5! \cdot 70!} = 17,259,390 \).

To find the probability of winning a lottery, we need to consider all possible outcomes. Here, it involves choosing 5 correct numbers out of 75 and a separate single number from 1 to 15. After calculating the combinations of choosing 5 numbers (17,259,390 ways), we find the ways to choose the additional number (15 ways). The total number of outcomes in this lottery becomes: \( 17,259,390 \times 15 = 258,890,850 \). Finally, the probability of winning the lottery is the inverse of the total number of possible outcomes: \( \frac{1}{258,890,850} \).

Comparing probabilities can provide insight into how likely different events are. In our example, we compare the lottery probability to the probability of being struck by lightning. The probability of being struck by lightning in a year is typically \( \frac{1}{960,000} \), which is much higher than the lottery probability of \( \frac{1}{258,890,850} \). This means you are far more likely to be struck by lightning than to win the jackpot. While both probabilities are small, the difference illustrates how some events are extraordinarily unlikely compared to others. Understanding these comparisons aids in comprehending risk in everyday life.

Calculating probability involves determining how many favorable outcomes exist versus the total number of possible outcomes. The general formula is: \( \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} \). For the lottery, the favorable outcome is winning, which means selecting the correct set of numbers. The total possible outcomes were earlier computed as 258,890,850. Therefore, the probability of winning is simply: \( \frac{1}{258,890,850} \). This very small fraction shows just how unlikely it is to win. Remember, calculating combination and understanding the total outcomes are fundamental steps in determining probabilities for complex events.