To determine the number of different 4-person committees that can be made out of 15 students, let's use the Combination Formula.
[tex]nCr=\frac{n!}{r!(n-r)!}[/tex]
where n = the total number of choices and r = the total number of selections.
In the question, our n = 15 because there are 15 students to choose from. Our r = 4 because 4 selections will be made from 15 students.
Let's replace the variables in the formula with their corresponding numerical value.
[tex]_{15}C_4=\frac{15!}{4!(15-4)!}[/tex]
Then, solve. Here are the steps.
1. In the denominator, subtract 15 and 4 first.
[tex]_{15}C_4=\frac{15!}{4!11!}[/tex]
2. Expand 15! until 12 only and cancel 11! Expand 4! too.
[tex]_{15}C_4=\frac{15\times14\times13\times12}{4\times3\times2\times1}[/tex]
3. Multiply the numbers in the numerator and denominator.
[tex]_{15}C_4=\frac{32,760}{24}[/tex]
4. Divide the numerator by the denominator.
[tex]_{15}C_4=1,365[/tex]
Therefore, there are 1, 365 different ways of forming a 4-person committee out of 15 students.
