There are 10 people at a meeting and there are 4 different door prizes to give out. How many ways can we select 4 people to receive the door prizes?

To find the number of ways to select 4 people out of 10 to receive the door prizes, we can use the combination formula. The combination formula is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Where: - \( n \) is the total number of items to choose from (in this case, 10 people) - \( k \) is the number of items to choose (in this case, 4 people) - \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \) Substitute the values into the formula: \[ C(10, 4) = \frac{10!}{4!(10-4)!} \] Now, we can calculate the number of ways to select 4 people out of 10 to receive the door prizes. Simplify the expression by following steps: - step0: Solution: \(\frac{10!}{\left(4!\times \left(10-4\right)!\right)}\) - step1: Remove the parentheses: \(\frac{10!}{4!\times \left(10-4\right)!}\) - step2: Subtract the numbers: \(\frac{10!}{4!\times 6!}\) - step3: Expand the expression: \(\frac{10\times 9\times 8\times 7\times 6\times 5\times 4!}{4!\times 6!}\) - step4: Simplify: \(\frac{10\times 9\times 8\times 7\times 6\times 5}{6!}\) - step5: Multiply: \(\frac{151200}{6!}\) - step6: Calculate the value: \(\frac{151200}{720}\) - step7: Reduce the fraction: \(210\) There are 210 ways to select 4 people out of 10 to receive the door prizes.