91Ó°ÊÓ

a) In how many ways can the letters in UNUSUAL be arranged? b) For the arrangements in part (a), how many have all three U's together? c) How many of the arrangements in part (a) have no consecutive U's?

Show that for all integers \(n, r \geq 0\), if \(n+1>r\), then $$ P(n+1, r)=\left(\frac{n+1}{n+1-r}\right) P(n, r) . $$

Find the coefficient of \(w^{2} x^{2} y^{2} z^{2}\) in the expansion of (a) \((w+x+y+z+1)^{10}\); (b) \((2 w-x+3 y+z-2)^{12} ;\) and, (c) \((v+w-2 x+y+5 z+3)^{12}\).

How many different paths in the \(x y\) plane are there from \((0,0)\) to \((7,7)\) if a path proceeds one step at a time by going either one space to the right \((R)\) or one space upward \((U)\) ? How many such paths are there from \((2,7)\) to \((9,14)\) ? Can any general statement be made that incorporates these two results?

Determine the sum of all the coefficients in the expansions of a) \((x+y)^{3}\) b) \((x+y)^{10}\) c) \((x+y+z)^{10}\) d) \((w+x+y+z)^{5}\) e) \((2 s-3 t+5 u+6 v-11 w+3 x+2 y)^{10}\).

a) If \(n\) and \(r\) are positive integers with \(n \geq r\), how many solutions are there to $$ x_{1}+x_{2}+\cdots+x_{r}=n $$ where each \(x_{i}\) is a positive integer, for \(1 \leq i \leq r ?\) b) In how many ways can a positive integer \(n\) be written as a sum of \(r\) positive integer summands \((1 \leq r \leq n)\) if the order of the summands is relevant?

Determine the number of six-digit integers (no leading zeros) in which (a) no digit may be repeated; (b) digits may be repeated. Answer parts (a) and (b) with the extra condition that the six-digit integer is (i) even; (ii) divisible by 5 ; (iii) divisible by \(4 .\)

a) In how many ways can seven people be arranged about a circular table? b) If two of the people insist on sitting next to each other, how many arrangements are possible?