In how many ways can the...

Question

In how many ways can the letters of the word CAT be arranged to form three letter words, if you are permitted to repeat a letter any number of times.

Answer 1

This problem is short enough that a list of possibilities is practical and instructive.

 

This list puts the letters in alphabetical order so we can remember them better.  It is like a bicycle lock or an old gas pump or an odometer (you do wish to drive someday, right?) -- there, the positions rotate 0-9 then make the next higher position increase by 1.  With 3 digits and 10 possibilities for each (0-9), there are 10*10*10 numbers from 000 up to 999.

 

Here, there are 3 positions with 3 possible values (and replacemet [ multiple use ] is allowed.  So, 3*3*3 = 27.

 

Study the list to see how A-C-T repeats and you will see how this works.

 

A A A

A A C

A A T

 

A C A

A C C

A C T

 

A T A

A T C

A T T

 

= = = = = = = =

 

C A A

C A C
C A T
 
C C A
C C C
C C T

C T A
C T C
C T T
 
= = = = = = = =

T A A
T A C
T A T

T C A
T C C
T C T

T T A
T T C
T T T

 

Now, if you were playing Boggle or working a crossword puzzle and needed real 3-letter words (and, I guess if you were a poor Scrabble player), you might use a dictionary (MS Word can check a dictionary while you type on a computer).  That's not a math problem, though.