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Problem 10

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

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.
See the step by step solution

## 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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