How many different strings of 5 lower case letters are there? Assume that letters may repeat and the string must contain at least one letter a? I tried this: 5×26^5=5×26 ×26 ×26 ×26=2684880
$\begingroup$
$\endgroup$
Add a comment
|
$\begingroup$
$\endgroup$
$(26)^5 - (25)^5.$
First term is total # of 5 character strings, where each character has 26 choices.
Second term is total # of 5 character strings, where each character has only 25 choices, re no "a" is used.
The difference between the two is the # of 5 character strings with at least 1 "a".
answered Dec 12 '20 at 1:19
$\begingroup$
$\endgroup$
Break it into five sub-problems: exactly 1 $A$, exactly 2 $A$s, ..., exactly 5 $A$s.
One $A$: There are five places for the required $A$, and $25^4$ ways to fill the remaining four slots without the use of another $A$.
Two $A$s: There are ${5 \choose 2}$ ways to place the two $A$s, and $25^3$ ways to fill the remaining $s$ slots with letters other than $A$.
...
Five $A$s: There is ${5 \choose 5} = 1$ way to fill with all five $A$s.
Add these up.
answered Dec 12 '20 at 1:20
