91Ó°ÊÓ

Find the number of integers between 1 and 10,000 that are neither perfect squares nor perfect cubes.

A bakery sells chocolate, cinnamon, and plain doughnuts and at a particular time has 6 chocolate, 6 cinnamon, and 3 plain. If a box contains 12 doughnuts, how many different options are there for a box of doughnuts?

Determine the number of solutions of the equation \(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=14\) in positive integers \(x_{1}, x_{2}, x_{3}, x_{4}\) and \(x_{5}\) not exceeding \(5 .\)

Determine the number of permutations of \(\\{1,2, \ldots, 8\\}\) in which no even integer is in its natural position.

Determine the number of permutations of \(\\{1,2, \ldots, 8\\}\) in which exactly four integers are in their natural positions.

Determine a general formula for the number of permutations of the set \(\\{1,2, \ldots, n\\}\) in which exactly \(k\) integers are in their natural positions.

At a party, seven gentlemen check their hats. In how many ways can their hats be returned so that (a) no gentleman receives his own hat? (b) at least one of the gentlemen receives his own hat? (c) at least two of the gentlemen receive their own hats?

Use combinatorial reasoning to derive the identity $$\begin{aligned}n !=\left(\begin{array}{c} n \\\0\end{array}\right) & D_{n}+\left(\begin{array}{c} n \\\1\end{array}\right) D_{n-1}+\left(\begin{array}{c} n \\\2\end{array}\right) D_{n-2} \\\&+\cdots+\left(\begin{array}{c}n \\ n-1\end{array}\right) D_{1}+\left(\begin{array}{l}n \\\n\end{array}\right) D_{0} \end{aligned}$$ (Here, \(D_{0}\) is defined to be 1.)

Prove that \(D_{n}\) is an even number if and only if \(n\) is an odd number.

Count the permutations \(i_{1} i_{2} i_{3} i_{4} i_{5} i_{6}\) of \(\\{1,2,3,4,5,6\\}\), where \(i_{1} \neq 1,5 ; i_{3} \neq\) \(2,3,5 ; i_{4} \neq 4 ;\) and \(i_{6} \neq 5,6\).