In how many ways can 20 students out of a class of 32 be chosen to attend class on a late Thursday afternoon (and take notes for the others) if (a) Paul refuses to go to class? (b) Michelle insists on going? (c) Jim and Michelle insist on going? (d) either Jim or Michelle (or both) go to class? (e) just one of Jim and Michelle attend? (f) Paul and Michelle refuse to attend class together?

(a)

$C(31, 20)=\dbinom{31}{20}=\dfrac{31!}{20!(31-20)!}=84672315$

(b)

$C(31, 19)=\dbinom{31}{19}=\dfrac{31!}{19!(31-19)!}=141120525$

(c)

$C(30, 18)=\dbinom{30}{18}=\dfrac{30!}{18!(30-18)!}=86493225$

(d)

$C(32, 20)-C(30, 20)$

$=\dfrac{32!}{20!(32-20)!}-\dfrac{30!}{20!(30-20)!}$

$=225792840-30045015$

$=1957457825$

(e)

$C(32, 20)-C(30, 20)-C(30, 18)$

$=225792840-30045015-86493225$

$=109254600$

(f)

$C(32, 20)-C(30, 18)$

$=225792840-86493225=139299615$