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To solve any problem involving combinations, we first need to understand the combination formula. This formula allows us to calculate the number of ways we can choose a subset of items from a larger set of items. The combination formula is given by: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)In this equation:

• \(n\) is the total number of items.
• \(k\) is the number of items we want to choose.

The exclamation mark (!) stands for factorial, which means multiplying a series of descending natural numbers.For instance, to find the number of ways to choose 3 items from a set of 12 items, we use \(n = 12\) and \(k = 3\). Placing these values into the formula, we get:\(\binom{12}{3} = \frac{12!}{3!(12-3)!}\).Next, we simplify the expression by substituting the values of the factorials.

Factorials are essential for calculating combinations. A factorial, represented by an exclamation point (!), is the product of all positive integers up to a given number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) .In our exercise, we need to calculate factorials for 12 and 3, and simplify the expression: \(12! = 12 \times 11 \times 10 \times 9!\),\(3! = 3 \times 2 \times 1 = 6\), and \((12-3)! = 9!\).We notice that the \(9!\) terms cancel out each other, simplifying our calculation to: \(\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1}\).This cancellation reduces large numbers and makes the calculation easy to handle.

Finally, let's perform the actual calculations to find our combination. After cancelling out the factorial terms, our simplified combination formula is: \(\binom{12}{3} = \frac{12 \times 11 \times 10}{6}\).First, calculate the numerator: \(12 \times 11 \times 10 = 1320\).Next, calculate the denominator: \(3 \times 2 \times 1 = 6\).Now divide the numerator by the denominator: \(1320 \div 6 = 220\).So, the number of ways to choose 3 items from 12 items is 220.Understanding probability calculations helps in solving various problems in statistics and combinatorics, making these concepts very important.