Prove that for each natural number n > 43, we can write
n = 6xn + 9yn + 20zn

for some nonnegative integers xn, yn, zn. then prove that 43 cannot be written in this
form.
(hint: write 44, 45, 46, 47, 48, and 49 in the given form. use induction to prove
that any larger number can be written in the given form.)

Aclassroom is made up of 11 boys and 14 girls. the teacher has four main classroom responsibilities that she wants to hand out to four different students (one for each of the four students). if the teacher chooses 4 of the students at random, then what is the probability that the four students chosen to complete the responsibilities will be all boys?