The circuit shown below uses TTL flip-flops. The flip-flops are triggered at the negative transitions of the clock. It is desired that when M = 1 the circuit should function as an up-counter (in 8421 BCD) and when M=0, as a down-counter. Design the combinational circuit interposed between the flip-flops so that the circuit works as desired. (i.e. find F as a function of Q,$$\overline Q $$, M, $$\overline M $$).

For the circuit shown below the output Fis given by

Minimum number of 2-input NAND gates required to implement the function, f=($$\overline X $$+$$\overline Y $$)(Z+W) is

The minimum number of 2-input NAND gates required to implement the Boolean function Z=A$$\overline {B\,} $$C, assuming that A, B and C are available, is