Given a pack of $52$ cards of classes hearts, spades, diamonds and clubs. Each of this class has an Ace, numbers $2,\space 3, \space \cdots, \space 10$, a Knave, a Queen and a King.
$5$ cards are selected successively returning each card after selection before selecting the next one.
What is the probability of having at least an Ace and at least a Knave?
My approach:
We are trying to choose $i$ Ace(s) and $j$ Knave(s) such that $1\le i,j \le 4$ and $i+j \le 5$ and then choose $(5-i-j)$ card(s) from the remaining $44$ cards. And the probability will be $\displaystyle \sum_{\substack{ 1\le i,j \le 4 \\ i+j\le 5 }}\cfrac{5!}{i!j!(5-i-j)!} \left(\cfrac 1{13}\right)^{i+j} \left(\cfrac {11}{13}\right)^{5-i-j}$ $ = \sum_{\substack{ 1\le i \le 4 \\ }}\sum_{\substack{ 1\le j \le 5-i \\ }}\cfrac{5!}{i!j!(5-i-j)!} \left(\cfrac 1{13}\right)^{i+j} \left(\cfrac {11}{13}\right)^{5-i-j}$
I am having problems computing this sum and I'm very sure that I'm missing something.