Number of ways of selecting 5 different letters:
$= {^5C_5} = 1$ way
Number of ways to select 2 similar and 3 different letter:
$={^4C_1} ×{^4C_3}=16$
Number of ways of selecting 2 similar + 2 more similar letter and 1 different letter:
$= {^4C_2}×{^3C_1}= 18$
Number of ways to select 3 similar and 2 different letter:
$= {^3C_1}×{^4C_2}= 18$
Number of ways to select 3 similar and another 2 other similar:
$= {^3C_1}×{^3C_1}=9$
Number of ways to select 4 similar and 1 different letter:
$= {^2C_1}×{^4C_1}=8$
ways of selecting 5 similar letters:
$= 1$
total ways: $= 1+16+18+18+9+8+1= \textbf{71}$
Alternately
A more detailed solution is as follows:
Table below can be scrolled horizontally

5 A’s
A A A A A 
4 B’s
B B B B 
3 C’s
C C C 
2 D’s
D D 
1 E
E 
Sub
Total 

SC1
5 same
0 diff.

${^1C_1}$
$=1$

NA

NA

NA

NA

1

SC2
4 same
1 diff.

$1×{^4C_1}$
$=4$

$1×{^4C_1}$
$=4$

NA

NA

NA

8

SC3
3 same
2 diff. each

$1×{^4C_2}$
$=6$

$1×{^4C_2}$
$=6$

$1×{^4C_2}$
$=6$

NA

NA

18

SC4
3 same
2 other same

$1×{^3C_1}$
$=3$

$1×{^3C_1}$
$=3$

$1×{^3C_1}$
$=3$

NA

NA

9

SC5
2 same
3 diff. each

$1×{^4C_3}$
$=4$

$1×{^4C_3}$
$=4$

$1×{^4C_3}$
$=4$

$1×{^4C_3}$
$=4$

NA

16

SC6
2 same
2 other same
1 diff.

$1×{^3C_1}×{^3C_1}$
$=9$

$1×{^2C_1}×{^3C_1}$
$=6$

$1×{^1C_1}×{^3C_1}$
$=3$

NA

NA

18

SC7
all diff.





${^5C_5}$
$=1$

1

Sum =






71

The solution can be structured in 7 'selection conditions' (SC 1 to SC 7)
SC1:
All five letters are the same
Only possible for $A = 1$ letter
⇒ Number of ways:
$= {^1C_1} = 1$ way  (1)
SC2:
Only 4 Letters are the same (Only possible with $A$ and $B$) and one is different (1 out of 4 remaining, after $A$ or $B$ is selected)
⇒ Number of ways:
$= 1×{^4C_1} + 1×{^4C_1} = 8$ ways  (2)
(Same as ${^2C_1}×{^4C_1}= 8$)
SC3:
Only 3 Letters are the same (Possible with $A, B$ and $C$) and two are diff.( 2 out of 4 remaining, after $A$ or $B$ or $C$ is selected)
⇒ Number of ways:
$= 1×{^4C_2}+1×{^4C_2}+1×{^4C_2} = 18$ ways  (3)
(same as ${^3C_1}×{^4C_2} = 18$)
SC4:
3 Letters are the same (Possible with $A, B$ and $C$) and two others are same (Possible with $A, B, C$ and $D ⇒ 2$ letters of 1 letter type from 3 remaining letter types)
⇒ Number of ways:
$= 1×{^3C_1}+1×{^3C_1}+1×{^3C_1} = 9$ ways  (4)
(Same as ${^3C_1}×{^3C_1} = 9$)
SC5:
2 Letters are the same (Possible with $A, B, C$ and $D$) and three are diff (same logic as above)
⇒ Number of ways:
$= 1×{^4C_3}+1×{^4C_3}+1×{^4C_3}+1×{^4C_3} = 16$ ways  (5)
(same as ${^4C_1}×{^4C_3} = 16$)
SC6:
2 Letters are the same (Possible with $A, B, C$ and $D$) and 2 other Letters are the same (Possible with $A, B, C$ and $D$)and one different letter (same logic as above)
⇒ Number of ways:
$= 1×{^3C_1}×{^3C_1} +1×{^2C_1}×{^3C_1} +1×{^1C_1}×{^3C_1} = 18$ ways  (6)
(Same as ${^4C_2}×{^3C_1}= 18$)
[SC6 is a little tricky because we need to avoid repetition as shown above]
SC7:
All 5 Letters are diff (Possible only with $A, B, C, D$ and $E$)
⇒ Number of ways $= {^5C_5} = 1$ way  (7)
Adding the number of ways from the 7 SCs:
$= (1) + (2) + (3) + (4) + (5) + (6) + (7)$
$= 1+8+18+9+16+18+1$
$= \textbf{71}$