Inclusion-Exclusion

A school has $147$ third graders. The third grade teachers have planned a special treat for the last day of school and brought ice cream for their students. There are three flavors: mint chip, chocolate, and strawberry. Suppose that $60$ students like (at least) mint chip, $103$ like chocolate, $50$ like strawberry, $30$ like mint chip and strawberry, $40$ like mint chip and chocolate, $25$ like chocolate and strawberry…

How many positive integers less than or equal to $1000$ are divisible by none of $3$, $8$, and $25$?

How many nonnegative integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 32$ with $0\leq x_i\leq 10$ for $i=1,2,3,4$?

A professor is working with six undergraduate research students. He has $12$ topics that he would like these students to begin investigating. Since he has been working with Katie for several terms, he wants to ensure that she is given the most challenging topic (and possibly others). Each topic must be assigned to a single student. Subject to this, in how many ways can he assign the topics to his stu…

A careless payroll clerk is placing employees' paychecks into envelopes that have been pre-labeled. The envelopes are sealed before the clerk realizes he didn't match the names on the paychecks with the names on the envelopes. If there are seven employees, in how many ways could he have placed the paychecks into the envelopes so that exactly three employees receive the correct paycheck?