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Evaluate each number. $$ C(4,3) $$

Short Answer

Expert verified
The combination \(C(4,3)\) represents the number of ways to choose 3 items from a set of 4. Using the formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), we can plug in the values and simplify the expression to find \(C(4,3) = 4\). Thus, there are 4 possible ways to choose 3 items from a set of 4.
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Step 1: Understand the Combination Formula

The formula to find the combinations is given by: \[C(n, k) = \frac{n!}{k!(n-k)!}\] where - n is the total number of items in the set - k is the number of items we want to choose - ! represents the factorial of a number (e.g., 5! = 5 × 4 × 3 × 2 × 1) In our case, we want to evaluate \(C(4,3)\), so we will plug in n = 4 and k = 3 into the formula to find the answer.

Step 2: Plug in the Values and Calculate Factorials

Using the Combination Formula, we'll plug in the values of n and k into the formula: \[C(4,3) = \frac{4!}{3!(4-3)!}\] Now we need to calculate the factorials: - 4! = 4 × 3 × 2 × 1 = 24 - 3! = 3 × 2 × 1 = 6 - (4-3)! = 1! = 1

Step 3: Apply the Factorials to the Formula

Now that we have calculated the factorials, we'll insert them back into the formula: \[C(4,3) = \frac{24}{6 \times 1}\]

Step 4: Simplify the Expression

Now, we just need to simplify the expression: \[C(4,3) = \frac{24}{6} = 4\]

Step 5: Interpret the Result

The value of \(C(4,3)\) is 4. This means that there are 4 possible ways to choose 3 items from a set of 4.

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