In how many ways can a member of a hiring committee select 3 of 12 job applicants for further consideration?

To find the number of ways a member of a hiring committee can select 3 out of 12 job applicants for further consideration, 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 job applicants (12 in this case) - \( k \) is the number of job applicants to be selected (3 in this case) - \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \) Substitute the values into the formula: \[ C(12, 3) = \frac{12!}{3!(12-3)!} \] Now, we can calculate the number of ways the hiring committee can select 3 out of 12 job applicants for further consideration. Simplify the expression by following steps: - step0: Solution: \(\frac{12!}{\left(3!\times \left(12-3\right)!\right)}\) - step1: Remove the parentheses: \(\frac{12!}{3!\times \left(12-3\right)!}\) - step2: Subtract the numbers: \(\frac{12!}{3!\times 9!}\) - step3: Expand the expression: \(\frac{12\times 11\times 10\times 9!}{3!\times 9!}\) - step4: Simplify: \(\frac{12\times 11\times 10}{3!}\) - step5: Multiply: \(\frac{1320}{3!}\) - step6: Calculate the value: \(\frac{1320}{6}\) - step7: Reduce the fraction: \(220\) The hiring committee can select 3 out of 12 job applicants for further consideration in 220 different ways.