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Complements

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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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