Consider a 2-bit saturating up/down counter that performs the saturating up count when the input $P$ is 0 , and the saturating down count when $P$ is 1 . The Next State table of the counter is as shown. The counter is built as a synchronous sequential circuit using $D$ flip-flops.

Inpur	Current state		Next state
$$ P $$	$$ Q_1 $$	$$ Q_0 $$	$$ Q_1^{+} $$	$$ Q_0^{+} $$
$$ \begin{aligned} & 0 \\ & 0 \\ & 0 \\ & 0 \\ & 1 \\ & 1 \\ & 1 \\ & 1 \end{aligned} $$	$$ \begin{aligned} & 0 \\ & 0 \\ & 1 \\ & 1 \\ & 0 \\ & 0 \\ & 1 \\ & 1 \end{aligned} $$	$$ \begin{aligned} & 0 \\ & 1 \\ & 0 \\ & 1 \\ & 0 \\ & 1 \\ & 0 \\ & 1 \end{aligned} $$	$$ \begin{aligned} & 0 \\ & 1 \\ & 1 \\ & 1 \\ & 0 \\ & 0 \\ & 0 \\ & 1 \end{aligned} $$	$$ \begin{aligned} & 1 \\ & 0 \\ & 1 \\ & 1 \\ & 0 \\ & 0 \\ & 1 \\ & 0 \end{aligned} $$

Which one of the following options corresponds to the expressions for the inputs of the $D$ flip-flops, $D_1$ and $D_0$ ?

Consider a Boolean function $F$ with the following minterm expression:

$$ F(P, Q, R, S)=\Sigma m(1,2,3,4,5,7,10,12,13,14) $$

Which of the following options is/are the minimal sum-of-products expression(s) of $F$ ?

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is

Consider $4 \times 4$ matrices with their elements from $\{0,1\}$. The number of such matrices with even number of 1 s in every row and every column is