How to solve these types of permutation problems? One thing I observe that, the way we calculate $nPn$, doesn't actually work for the cases of $nPr$.
(1) How many permutations are possible if we take 3 letters at a time from the list {A,A,B,C,E} where the words always start with an A and ends with an A?
Ans:  $\frac{^{5-2}P_{3-2}}{2!} = 3$ .
(2) How many permutations are possible if we take 3 letters at a time from the list {A,A,B,C,E} where the words always start with an AA?
Ans:  $\frac{^{5-2}P_{3-2}}{2!} = 3$ .
(3) How many permutations are possible if we take 3 letters at a time from the list {A,A,B,C,E} where the words always start with an A?
Ans:  $^{5-1}P_{3-1} = 12$ .
(4) How many of them contain AA?
Answer: 3P2 = 6.
(5) How many permutations are possible if we take 3 letters at a time from the list {A,A,B,C,E} where the words always start with an E?
Ans: 
The answer should be 7. 
But, $^{5-1}P_{3-1} = 12$ is wrong.
(6) How many of them start with a vowel?
Ans: 
The answer should be 19. 
But, $^{5-3}P_{3-3} = 2$ is wrong.
My question is, which problems should I solve by using multiplications between different permuted values, which questions should I solve by using additions between different permuted values, and, which problems should I solve by hand-counting (ie. Permutation formula doesn't work)?
Another question is, how should I deal with overcounting?

The answer to 5 th is 7 . See  we have to select three lettets where first letter E is fixed. Now  from  4 letters we have to select two so we have  4C1 but A is common so divided by 2 and next letter can be selected in 3C1 ways . So answer is 1+(4C1/2.3C1)=7. Note  the difference between permutation and combination. Combination is selection of n things taken r at a time while permutation is arrangement of distinct things taken r at a time.
