Question

You are going to create a password consisting of 2 digits (0-9) followed by 4 letters from the alphabet (A-2); {such as “23ABCD', for example). a) How many total unique passwords can be created in this way if the digits and letters as described cannot be repeated? Show your work. b) How many total unique passwords can be created if the first digit must be a 0 or 1, the last letter must be an A, B, or C, and if digits and letters are allowed to be repeated in the other spots? Show your work,

Answer 1

a)

For letters:

$\begin{gathered} nCk=\frac{n!}{k!(n-k)!} \\ 26C4=\frac{26!}{4!(22)!}=14950 \end{gathered}$

For numbers:

$10C2=\frac{10!}{2!(8)!}=45$

The number of passwords is:

$14950\times45=672750$

b)

For the first and the last spots

$\begin{gathered} 2C1=2 \\ 3C1=3 \end{gathered}$

Since we can repeat for the other spots:

$10\times26\times26\times26=175760$

The number of passwords is:

$175760\times2\times3=1054560$