Question

A security system uses a 3-letter password, but no letter can be used more than once. How many possible passwords are there if 6 of the letters of the alphabet can be used by the system?

Answer 1

The correct answer is
`\left[\begin{array}{ccc}26\\6\end{array}\right]` ×
`\left[\begin{array}{ccc}6\\3\end{array}\right]`.

Explanation:

Total number of alphabets the system can use = 6

In English language there are 26 alphabets and we are to use any 6 of them. Thus this can be done in
`\left[\begin{array}{ccc}26\\6\end{array}\right]` ways.

A password is to be made for a security system using the above 6 letters and no letter can be used more than once.

Thus number of possible ways of making passwords of three letters using the above six letters are
`\left[\begin{array}{ccc}6\\3\end{array}\right]`.

Total number of possible ways to choose 6 alphabets and make a password of 3 letters are
`\left[\begin{array}{ccc}26\\6\end{array}\right]` ×
`\left[\begin{array}{ccc}6\\3\end{array}\right]`.