Question

Define permutations and combinations
A. Determine 7P3
B. Determine 9P3
C. How many types of answers do the students of the 7 questions given should only be answered 5?

Answer 1

$\displaystyle ^7\text{P}_3=\frac{7!}{(7-3)!}\\^7\text{P}_3=\frac{7!}{4!}\\^7\text{P}_3=\frac{7\cdot6\cdot5\cdot4!}{4!}\\^7\text{P}_3=7\cdot6\cdot5\\\boxed{\boxed{^7\text{P}_3=210}}\\\\^9\text{P}_3=\frac{9!}{(9-3)!}\\^9\text{P}_3=\frac{9!}{6!}\\^9\text{P}_3=\frac{9\cdot8\cdot7\cdot6!}{6!}\\^9\text{P}_3=9\cdot8\cdot7\\\boxed{\boxed{^9\text{P}_3=504}}\\\\n={^7\text{C}_5}\\n=\frac{7!}{(7-5)!5!}\\n=\frac{7\cdot6\cdot5!}{2!5!}\\n=\frac{7\cdot6}{2}\\\boxed{\boxed{n=21}}$