Question 1203679
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1-letter words....
There are 4 different letters to choose from.
The number of 1-letter words is 4.<br>
2-letter words....
If both letters A are used, there is only 1 word -- AA
If the letter A is not repeated, then there are 4 choices for the first letter and then 3 choices for the second, so there are 4*3=12 2-letter words without both letters A.
The number of 2-letter words is 1+12 =13.<br>
3-letter words....
If both letters A are used, then the combination of letters can be AAI, AAR, or AAT.  There are (3!)/(2!) = 3 arrangements of each of those, for a total of 9.
If both letters A are not used, then there are 4 choices for the first letter, 3 for the second, and 2 for the third, for a total of 4*3*2 = 24.
The number of 3-letter words is 9+24 = 33.<br>
4-letter words....
If both letters A are used, then the combination of letters can be AAIR, AAIT, or AART.  There are (4!)/(2!) = 12 arrangements of each of those, for a total of 36.
If both letters A are not used, then the letters are ATRI in any order; there are 4! = 24 of those.
The number of 4-letter words is 36+24 = 60.<br>
5-letter words....
The letters are ATARI in any order, the number of arrangements is (5!)/(2!) = 60.
There are 60 5-letter words.<br>
Summary:
1-letter words: 4
2-letter words: 13
3-letter words: 33
4-letter words: 60
5-letter words: 60<br>
Total number of words: 4+13+33+60+60 = 170<br>
ANSWERS:
A) 5-letter words: 60
B) 2-letter words: 13
C) total number of words: 170<br>