Solve the following two " union " type questions: (a) How many bit strings of length 8 either begin with 2 0s or end with 1 1s? (inclusive or) (b) Every student in a discrete math class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 36 computer science majors (including joint majors), 29 math majors (including joint majors) and 5 joint majors?

a) Let set A be the bit strings of length 8 beginning with two 0s:

$|A|=1\cdot1\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2=2^6=64$

Let set B be the bit strings of length 8 ending with one 1s:

$|B|=2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot1=2^7=128$

Let C be the bit strings of length 8 beginning with two 0s and ending with one 1s:

$|C|=|A\cap B|=1\cdot1\cdot2\cdot2\cdot2\cdot2\cdot2\cdot1=2^5=32$

Then

$|A\cup B|=|A| + |B|-|A\cap B|=64+128-32=160$

b) Let A be computer science majors

$|A|=36$

Let set B be math majors

$|B|=29$

And joint majors are $|A \cap B|= 5$

Therefore there are students in the class:

$|A \cup B|=|A|+|B|-|A \cap B|=36+29-5=60$

Answer:

a) 160

b) 60