Probability

The problem at hand is to calculate the combination of 12 items taken 4 at a time, often denoted as 12C4. This is a common problem in combinatorics, a branch of mathematics that deals with counting and arranging objects. Step 1: Understand the Combination Formula The formula for combinations is given by: C(n, r) = n! / [r!(n-r)!] where: - n is the total number of items, - r is the number of items to choose, - "!" denotes factorial, which is the product of all positive integers up to that number. Step 2: Plug in the Values In our case, n = 12 and r = 4. So, we need to calculate: C(12, 4) = 12! / [4!(12-4)!] Step 3: Simplify the Factorials First, calculate the factorials: 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 4! = 4 × 3 × 2 × 1 (12-4)! = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 Step 4: Substitute the Factorials into the Formula C(12, 4) = (12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / [(4 × 3 × 2 × 1) × (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)] Step 5: Simplify the Expression Notice that the terms in the denominator (4!, 8!) are also in the numerator as part of 12!. This allows us to cancel out these terms: C(12, 4) = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1) Step 6: Calculate the Result Finally, perform the multiplication and division to get the result: C(12, 4) = 495 So, there are 495 ways to choose 4 items from a set of 12.

Question

Calculate 12c4.

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