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Consider a hypothetical control unit that supports 5 groups of mutually exclusive control signals. Also assume that group 1 & group 2 are using horizontal micro-programming whereas group-3, 4 and 5 are using vertical micro-programming. The total number of bits used for control words are ________.     My doubt is what does it mean by 5 groups of mutually exclusive signals?

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Here it says G1 and G2 are of 43 and 29 signals each and are horizontally microprogrammed words. So $43 + 29$ bits.
G3, G4 and G5 are are vertically microprogrammed and should be able to generate 76, 130 and 100 signals each. So that is $log_276 + log_2130 + log_2100$.
$43+29+\left \lceil log_276 \right \rceil + \left \lceil log_2130 \right \rceil + \left \lceil log_2100 \right \rceil$ to be precise ;)