ANSWER
[tex]\begin{gathered} A)2176782336 \\ B)1402410240 \end{gathered}[/tex]EXPLANATION
A) We want to find how many 6 character passwords are possible if characters can be repeated.
There are 26 letters of the alphabet and 10 numeric digits (0 - 9).
This means that each number slot has 36 choices that can be made.
Therefore, if the characters can be repeated, it means that the number of 6 character passwords possible is:
[tex]\begin{gathered} 36^6 \\ \Rightarrow2176782336 \end{gathered}[/tex]B) There are still 36 choices possible but this time there will not be any repetition.
To find the number of possible 6 character passwords, we apply the permutation formula:
[tex]^nP_r=\frac{n!}{(n-r)!}[/tex]Therefore, we have:
[tex]\begin{gathered} ^{36}P_6=\frac{36!}{(36-6)!}=\frac{36!}{30!} \\ \Rightarrow\frac{36\cdot35\cdot34\cdot33\cdot32\cdot31\cdot30!}{30!} \\ \Rightarrow36\cdot35\cdot34\cdot33\cdot32\cdot31 \\ \Rightarrow1402410240 \end{gathered}[/tex]That is the answer.
