algorithm for number of subsequences containing at most k numbers with no element repeated in each of the subsequence
You can solve this using dynamic programming. Let me show instead the solution using generating functions, which can also be implemented using dynamic programming.

Suppose that there are $m$ distinct values in the array, appearing $n_1,\ldots,n_m$ times. The generating function
$$ \prod_{i=1}^m (1+n_ix) $$
counts the number of subsequences with distinct values of size $k$, in the sense that this is the coefficient of $x^k$. For example, in your case $m = 3$ and $n_1=n_2=2,n_3=1$. Therefore the generating function is
$$
(1+2x)^2(1+x) = 
(1+4x+4x^2)(1+x) =
1+5x+8x^2+4x^3.
$$