The number of min-terms after minimizing the following Boolean expression is _______________ . $$$\left[ {D' + AB' + A'C + AC'D + A'C'D} \right]'$$$

A half adder is implemented with $$XOR$$ and $$AND$$ gates. A full adder is implemented with two half adders and one $$OR$$ gate. The propagation delay of an $$XOR$$ gate is twice that of an $$AND/OR$$ gate. The propagation delay of an $$AND/OR$$ gate is $$1.2$$ microseconds. A $$4$$-bit ripple-carry binary adder is implemented by using four full adders. The total propagation time of this $$4$$-bit binary adder in microseconds is____________ .

The minimum number of $$JK$$ flip-flops required to construct a synchronous counter with the count sequence $$\left( {0,0,1,1,2,2,3,3,0,0,...} \right)$$ is ____________.

Let $$X$$ and $$Y$$ denote the sets containing $$2$$ and $$20$$ distinct objects respectively and $$𝐹$$ denote the set of all possible functions defined from $$X$$ to $$Y$$. Let $$f$$ be randomly chosen from $$F.$$ The probability of $$f$$ being one-to-one is ________.