Question

5 there are 18 students on the debate club. how many ways can the club send 4 students to the next debate? 6 how many 5 - character passwords can be created using the letters from the word pennsylvania if the letters in the password can not repeat? 7 if sixteen people tried out for the lead in a musical, how many ways can the director select the lead and an alternate? 8 ava is using beads to make a bracelet. how many ways can she arrange the first 7 beads if she has 12 different colors to choose from and colors cannot repeat?

Explanation:

Step1: Recall permutation formula

The permutation formula is $P(n,r)=\frac{n!}{(n - r)!}$, where $n$ is the total number of items and $r$ is the number of items to be selected.

Step2: Solve problem 5

For problem 5, $n = 18$, $r=4$. So $P(18,4)=\frac{18!}{(18 - 4)!}=\frac{18!}{14!}=18\times17\times16\times15 = 73440$.

Step3: Solve problem 6

The word "PENNSYLVANIA" has 12 distinct - like letters. For a 5 - character password, $n = 12$, $r = 5$. So $P(12,5)=\frac{12!}{(12 - 5)!}=\frac{12!}{7!}=12\times11\times10\times9\times8=95040$.

Step4: Solve problem 7

For selecting a lead and an alternate from 16 people, $n = 16$, $r = 2$. So $P(16,2)=\frac{16!}{(16 - 2)!}=\frac{16!}{14!}=16\times15 = 240$.

Step5: Solve problem 8

For arranging 7 beads out of 12 different colors, $n = 12$, $r = 7$. So $P(12,7)=\frac{12!}{(12 - 7)!}=\frac{12!}{5!}=12\times11\times10\times9\times8\times7\times6=3991680$.

Answer:

Problem 5: 73440
Problem 6: 95040
Problem 7: 240
Problem 8: 3991680