We'll break our set of letters in $2$ parts
$1^{st}$ part: (contains 2 set)
one set will be having 2 E's i.e $S_1 = \{E,E\}$ & another one $S_2$ will be having $\{K, P, L, R\}$
and $2^{nd}$ part:
this part contains only one set $S_3 = \{K,P,L,R,E\}$
First, we'll start with $1$- letter words
The total no. of distinct words possible of length $1 \Rightarrow 5 \text{ (as there are 5 distinct words)}$
Now, $2$ letter- words
Here $2$ cases are possible :
1) taking $2$ E from $S_1$ (can be done in $1$ ways only) & taking no letter from $S_2$ & then arranging them(can be done in $1$ ways as both are identical)
OR
2) taking $2$ letters from $S_3$ & then arranging them which can be done in $^5C_2 \times 2! = ^5P_2$
$\therefore$ The total no. of distinct words possible of length $2 \Rightarrow (1 \times 1) + ^5P_2 = 1+20 = 21$
$3-$letter words:
$2$ cases are possible:
1) Choosing 2 place from the 3 places(can be done in $^3C_2 ways$) & then taking $2$ letters from $S_1$(can be done in $1$ ways) & $1$ letter from $S_2$ (can be done in $^4C_1$ ).
This can be done in $^3C_2 \times 1 \times ^4C_1 \\ = 3 \times 4 = 12 \text{ ways}$
OR
2) Taking $3$ letters from $S_3$ & arranging them, which can be done in $^5P_3 = 60 \hspace{0.1cm} ways$
$\therefore$ The total no. of distinct words possible of length $3 \Rightarrow 12 + 60 = 72$
$4-$ letter words:
Again $2$ cases are possible:
1) Choosing $2$ positions from the $4$ positions (can be done in $^4C_2$ ways) & then taking $2$ letters from $S_1$ (can be done in $1$ ways) and remaining $2$ letters we can choose from $S_2$ & arrange them (can be done in $^4P_2 = 12$ ways )
This can be done in $^4C_2 \times 1 \times 12 \\ = 6 \times 12 \\ = 72 \text{ ways}$
2) Choosing all the $4$ letters from $S_3$ & arrange them, which can be done in $^5P_4 = 120$ ways.
$\therefore$ The total no. of distinct words possible of length $4 \Rightarrow 72 + 120 = 192$
$5-$ letter words:
$2$ cases are possible again :
1) Choosing $2$ positions from the $5$ positions (can be done in $^5C_2$ ways) & then taking $2$ letters from $S_1$ (can be done in $1$ ways) and remaining $3$ letters we can choose from $S_2$ & arrange them (can be done in $^4P_3 = 24$ ways )
This can be done in $^5C_2 \times 1 \times 24 \\ = 10 \times 24 \\ = 240 \text{ ways}$
2) Choosing all the $5$ letters from $S_3$ & arrange them, which can be done in $^5P_5 = 120$ ways.
$\therefore$ The total no. of distinct words possible of length $4 \Rightarrow 240 + 120 = 360$
$6-$ letter words:
Now, we'll be taking all $6$ words from our original letter set i.e. $\{K, E, P, L, E, R\}$
So, the total no. of distinct words possible of length $6$ will be $\Rightarrow \dfrac{^6P_6}{2!} = 360$
$\therefore$ The no. of distinct words of any (nonzero) length can be formed using the letters of $KEPLER$
at most once each will be $= 5 + 21 + 72 + 192 + 360 + 360 \\ = 1010 $
