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To determine the number of different simple random samples, you need the combination formula. This formula is essential for calculating how many ways you can choose a sample of items from a larger set without regard to the order. The combination formula is represented as: i) \( C(n, k) = \frac{n!}{k!(n-k)!} \) Where: n) \(n\) is the population size k) \(k\) is the sample size ii) For our case, you have a population size of 100, and you need a sample size of 7. Hence, \(C(100, 7)\) is what you are solving. When using the formula, it is also important to substitute the values correctly. This helps in getting the accurate count of different possible samples.

Factorials play a crucial role in computing combinations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a certain number. For example: i) The factorial of 5, written as \(5!\), is calculated as: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] ii) To solve \(C(100, 7)\), you will be calculating three factorials: \(100!\), \(7!\), and \(93!\). \(100!\) is the multiplication of all numbers from 100 down to 1, which is a massive number. Factorials grow rapidly with larger numbers and are best computed using a calculator or specific software. Remember, understanding factorials makes it easier to handle problems involving combinations.

The size of the sample chosen represents the number of items you want to pick out from a larger population. For simple random sampling: i) Each item in the population has an equal chance of being selected. ii) In our example, the sample size (\(k\)) is 7, and the population size (\(n\)) is 100. It is important to decide on the sample size carefully to ensure that the sample accurately represents the population. The sample size used influences the precision of your results. Simple random sampling can be depicted as drawing 7 items from a bag containing 100 items, where each selection is independent of the others.