Probability

# Relative Complement

The student council of Ray School consists of $6$ seniors, $5$ juniors, and $3$ sophomores. How many ways are there to form a delegation of $4$ representatives for a school board meeting if the delegation must consist only of juniors and seniors and must include at least $1$ junior?

Tina has chosen some books at the book store, and she would like to buy: $6$ novels, $6$ math books, and $3$ biographies. However, Tina can only afford to buy $4$ books.

Since she can't buy them all, if she needs at least $1$ math book and puts back all the biographies, how many difference choices of books could she make?

Ralph chooses $4$ numbers among $4, 5, 6$ and $7$, with duplication allowed, to create a $4$-digit number larger than $6000$. How many ways can he obtain a number such that at least one of the middle two digits is $7?$

In a town hall meeting, $100$ people voted on two issues $a$ and $b.$ The results show that $58$ people voted in favor of $a$ and the rest voted against it, and $68$ people voted in favor of $b$ and the rest voted against it. Also, the number of people who voted against both $a$ and $b$ is greater by $4$ than $\frac{1}{3}$ of the number of people who voted in favor of both $a$ and $b.$ How many people voted in favor of both $a$ and $b?$

Liza picks $4$ distinct numbers from $1, 2, 3, 4, 5$ to create a $4$-digit number. How many ways are there to form a number such that either the first or last digit (or both) is even?

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