Question

question 7
3 pts
the u.s. olympic rowing coach has been given a list of 12 individuals to form the usa mens eight rowing team, which should have 8 members. how many different teams of 8 may be selected from the list of individuals?
312
380
the correct answer is not listed
603
506
178
290
495

Explanation:

Step1: Use combination formula

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 12\) (total number of individuals) and \(r=8\) (number of team - members to be selected).

Step2: Calculate factorial values

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So, \(12! = 12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1\), \(8! = 8\times7\times6\times5\times4\times3\times2\times1\), and \((12 - 8)!=4! = 4\times3\times2\times1\).
Then \(C(12,8)=\frac{12!}{8!×4!}=\frac{12\times11\times10\times9\times8!}{8!\times4\times3\times2\times1}\).

Step3: Simplify the expression

Cancel out the \(8!\) terms. We get \(\frac{12\times11\times10\times9}{4\times3\times2\times1}\).
\(12\div4 = 3\), \(10\div2=5\). The expression becomes \(\frac{3\times11\times5\times9}{3\times1}\). Cancel out the \(3\) terms. Then \(11\times5\times9=495\).

Answer:

495