91Ó°ÊÓ

Consider the \(2^{19}\) compositions of 20 . (a) How many have each summand even? (b) How many have each summand a multiple of 4 ?

a) Find the number of ways to write 17 as a sum of \(1^{\prime} s\) and 2 's if order is relevant. b) Answer part (a) for 18 in place of 17 . c) Generalize the results in parts (a) and (b) for \(n\) odd and for \(n\) even.

Find the value(s) of \(n\) in each of the following: (a) \(P(n, 2)=90\), (b) \(P(n, 3)=3 P(n, 2)\), and (c) \(2 P(n, 2)+50=P(2 n, 2)\).

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?

a) How many distinct paths are there from \((-1,2,0)\) to \((1,3,7)\) in Euclidean three-space if each move is one of the following types? \((\mathrm{H}):(x, y, z) \rightarrow(x+1, y, z)\) \((\mathrm{V}):(x, y, z) \rightarrow(x, y+1, z)\) \((\mathrm{A}):(x, y, z) \rightarrow(x, y, z+1)\) b) How many such paths are there from \((1,0,5)\) to \((8,1,7) ?\) c) Generalize the results in parts (a) and (b).

For any positive integer \(n\) determine a) \(\sum_{i=0}^{n} \frac{1}{i !(n-i) !}\) b) \(\sum_{i=0}^{n} \frac{(-1)^{t}}{i !(n-i) !}\)

A sequence of letters of the form abcba, where the expression is unchanged upon reversing order, is an example of a palindrome (of five letters). (a) If a letter may appear more than twice, how many palindromes of five letters are there? of six letters? (b) Repeat part (a) under the condition that no letter appears more than twice.

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 possible ways could a student answer a 10-question true-false test? b) In how many ways can the student answer the test in part (a) if it is possible to leave a question unanswered in order to avoid an extra penalty for a wrong answer?