91Ó°ÊÓ

How many different linear arrangements are there of the letters \(A, B, C, D\), E, \(F\) for which (a) \(\mathrm{A}\) and \(\mathrm{B}\) are next to each other; (b) A is before \(B\); (c) \(\mathrm{A}\) is before \(\mathrm{B}\) and \(\mathrm{B}\) is before \(\mathrm{C}\); (d) \(\mathrm{A}\) is before \(\mathrm{B}\) and \(\mathrm{C}\) is before \(\mathrm{D}\); (e) \(\mathrm{A}\) and \(\mathrm{B}\) are next to each other and \(\mathrm{C}\) and \(\mathrm{D}\) are also next to each other; (f) \(\mathrm{E}\) is not last in line?

How many outcome sequences are possible when a die is rolled four times, where we say, for instance, that the outcome is \(3,4,3,1\) if the first roll landed on 3 , the second on 4 , the third on 3 , and the fourth on \(1 ?\)

A president, treasurer, and secretary, all different, are to be chosen from a club consisting of 10 people. How many different choices of officers are possible if (a) there are no restrictions; (b) \(A\) and \(B\) will not serve together; (c) \(C\) and \(D\) will serve together or not at all; (d) \(E\) must be an officer; (e) \(F\) will serve only if he is president?

In how many ways can \(r\) objects be selected from a set of \(n\) if the order of selection is considered relevant?

There are \(\left(\begin{array}{l}n \\ r\end{array}\right)\) different linear arrangements of \(n\) balls of which \(r\) are black and \(n-r\) are white. Give a combinatorial explanation of this fact.

A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many if she must answer at least 3 of the first 5 questions?

John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If e?ch of the boys can play all 4 instruments, how many different arrangements are possible? What if John and Jim can play all 4 instruments, but Jay and Iack can each nlay only niano and drums?

In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each?

How many different 7-place license plates are possible when 3 of the entries are letters and 4 are digits? Assume that repetition of letters and numbers is allowed and that there is no restriction on where the letters or numbers can be placed.

Give a combinatorial explanation of the identity $$ \left(\begin{array}{l} n \\ r \end{array}\right)=\left(\begin{array}{c} n \\ n-r \end{array}\right) $$