Letter combinations

$\begin{matrix} All\quad alike & ^{ 1 }{ { C }_{ 1 } } & 1 \\ 4\quad alike,\quad 1diff & ^{ 2 }{ { C }_{ 1 } }\times ^{ 4 }{ { C }_{ 1 } } & 8 \\ 3\quad alike,\quad 2\quad alike & ^{ 3 }{ { C }_{ 1 } }\times ^{ 3 }{ { C }_{ 1 } } & 9 \\ 3\quad alike,\quad 2\quad diff & ^{ 3 }{ { C }_{ 1 } }\times ^{ 4 }{ { C }_{ 2 } } & 18 \\ 2\quad alike,\quad 2alike,\quad 1diff & ^{ 4 }{ { C }_{ 2 } }\times ^{ 3 }{ { C }_{ 1 } } & 18 \\ 2\quad alike,\quad 3\quad diff & ^{ 4 }{ { C }_{ 1 } }\times ^{ 4 }{ { C }_{ 3 } } & 16 \\ All\quad diff & ^{ 5 }{ { C }_{ 5 } } & 1 \\ \quad & \quad & 71 \end{matrix}$

If casework isn't your strongest area, here is a generating functions approach. Let $x_i, \ i=1 \ \text{to} \ 5,$ be the number of A's, B's, C's, D's and E's selected respectively.

Now, note that $\displaystyle\sum_{i=1}^{5} x_i=5$ . Also note that $0 \leq x_i \leq 6-i$ .

Therefore the required number of ways

$\begin{aligned} &= \text{Coefficient of} \ x^5 \ \text{in} \ \ \prod_{q=1}^{5} \sum_{p=0}^{q} x^p\\ &\equiv \text{Coefficient of} \ x^5 \ \text{in} \ \ (1-x^6)(1-x^5)(1-x^4)(1-x^3)(1-x^2)(1-x)^{-5}\\ &\equiv \text{Coefficient of} \ x^5 \ \text{in} \ \ (1-x^4-x^5)(1-x^2-x^3+x^5)(1-x)^{-5}\\ &\equiv \text{Coefficient of} \ x^5 \ \text{in} \ \ (1-x^2-x^3-x^4)(1-x)^{-5}\\ &=\binom{9}{5}-\binom{7}{3}-\binom{6}{2}-\binom{5}{1}=\boxed{71} \end{aligned}$

Note 1 : I have neglect terms with powers of $x>5$ .

Note 2 : $(1-x)^{-5}=\displaystyle\sum_{r=0}^{\infty} \binom{4+r}{r} x^r$

Dynamic programming (on paper): let $P[i][j]$ be the number of ways to take $j$ letters out of the first $i$ . Clearly, for $j < 0$ , $P[i][j]=0$ , and for $i=0$ , $P[0][0]=1$ and all other $P[0][j]=0$ .

Then, if you have $D[i]$ letters of type $i$ , you can calculate $P[i][j]=\sum_{k=j-D[i]}^j{P[i-1][k]}$ on paper for all $j \le 5$ ; the answer is $P[5][5]$ .