Answer
$450$ ways
Work Step by Step
The examine has a sequence of selections:
1. ... 8 out of a group of 10 multiple choice questions ... in ${}_{10}C_{8}$ ways
2. ... 3 out of a group of 5 open-ended problems... in ${}_{5}C_{3}$ ways
By the Fundamental Counting Principle,
Total ways= ${}_{10}C_{8}\cdot {}_{5}C_{3}$
${}_{10}C_{8}=\displaystyle \frac{10!}{(10-8)!8!}=\frac{10\times 9}{1\times 2}=45$
${}_{5}C_{3}=\displaystyle \frac{5!}{(5-3)!3!}=\frac{5\times 4}{1\times 2}=10$
Total ways= ${}_{10}C_{8}\cdot {}_{5}C_{3} =45\times 10=450$
