SOLUTION: How many five-letter words are there in which no letter appears 3 or more times in a row? a. compute the answer by using an indirect counting method b. check your answer to part

Indirectly:

There are 265 ways to arrange 5 alphabet letters in a row

That's 11881376 ways

We need to subtract the unacceptab;le ones:

1. Those which have exactly 5 letters in a row all the same

12345
XXXXX

There are 26 ways to choose what X represents.
That's 26 ways

That's 26 ways to subtract

2. Those which have exactly 4 letters in a row the same:

12345  12345  
XXXX-  -XXXX

Choose the position for the four in a row in 2 ways
Choose the X 26 ways
Choose the - 25 ways

That's 2x26x25 = 1300 more ways to subtract


3. Those which have exactly 3 letters in a row on either end:

12345  12345
XXX--  --XXX 

Choose the position for the three in a row in 2 ways
Choose the X 26 ways
Choose the - that's next to an X in 25 ways
Choose the - on the end in 26 ways (it can be the same as X or its neighbor)

That's 2x26x25x26 = 33800 ways to subtract

4. Those which have exactly 3 letters in a row in the middle:

12345
-XXX- 

Choose the X 26 ways
Choose the left - in 25 ways
Choose the right - in 25 ways 

That's 26x25x25 = 16250 more ways to subtract

Total = 11881376 - 26 - 1300 - 33800 - 16250 = 11830000

------------------------------

Direct.

1.  Those which have no two in a row:

12345
-----

Choose letter #1 26 ways
Choose letter #2 25 ways
Choose letter #3 25 ways (can be the same as 1 but not 2
Choose letter #4 25 ways (can be the same as 1 or 2 but not 3
Choose letter #5 25 ways (can be the same as 1 or 2 or 3 but not 4  

That's 26x25x25x25x25 = 10156250

2.  Those which have exactly 1 pair of letters in a row.

(a) 12345   (b) 12345   (c) 12345   (d) 12345
    XX---       -XX--       --XX-       ---XX

Choose the position for the 1 pair in a row in 4 ways
Choose the letter for X in 26 ways
Choose letter #3 in (a), letter #4 in (b), letter #2 in (c) 
and letter #3 in (d) in 25 ways
Choose letter #4 in (a), letter #5 in (b), letter #1 in (c) 
and letter #2 in (d) in 25 ways
Choose letter #5 in (a), letter #1 in (b), letter #5 in (c) 
and letter #1 in (d) in 25 ways

That's 4x26x25x25x25 = 1625000

3.  Those which have exactly 2 pairs of letters in a row
on either end.

12345  12345  
XXYY-   XXYY 

Choose the position for the 2 pair in a row in 2 ways
Choose the letter for X in 26 ways
Choose the letter for Y in 25 ways
Choose the - in 25 ways

That's 2x26x25x25 = 32500

4.  Those which have two pairs on each end:

12345   
XX-YY   

Choose the letter for X in 26 ways
Choose the - in 25 ways
Choose the letter for Y in 25 ways

That's 26x25x25 = 16250

Total = 10156250 + 1625000 + 32500 + 16250 =  11830000

Edwin