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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

(a) Provide a combinatorial argument to show that if n and k are positive integers with n = 3k, then n!/(3!)k is an integer. (b) Generalize the result of part (a).
How many distinct four-digit integers can one make from the digits 1, 3, 3, 7, 7, and 8? 35
(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?
(a) In how many ways can eight people, denoted A, B, . . . , H be seated about the square table shown in Fig. 1.6, where Figs. 1.6(a) and 1.6(b) are considered the same but are distinct from Fig. 1.6(c)?(b) If two of the eight people, say A and B, do not get along well, how many different seating's
Sixteen people are to be seated at two circular tables, one of which seats 10 while the other seats six. How many different seating arrangements are possible?
A committee of 15 - nine women and six men - is to be seated at a circular table (with 15 seats). In how many ways can the seats be assigned so that no two men are seated next to each other?
Write a computer program (or develop an algorithm) to determine whether there is a three digit integer abc (= 100a + 10b + c) where abc = a! + b! + c!.
The board of directors of a pharmaceutical corporation has 10 members. An upcoming stockholders' meeting is scheduled to approve a new slate of company officers (chosen from the 10 board members). (a) How many different slates consisting of a president, vice president, secretary, and treasurer can
While on a Saturday shopping spree Jennifer and Tiffany witnessed two men driving away from the front of a jewelry shop, just before a burglar alarm started to sound. Although everything happened rather quickly, when the two young ladies were questioned they were able to give the police the
To raise money for a new municipal pool, the chamber of commerce in a certain city sponsors a race. Each participant pays a $5 entrance fee and has a chance to win one of the different-sized trophies that are to be awarded to the first eight runners who finish. (a) If 30 people enter the race, in
Matthew works as a computer operator at a small university. One evening he finds that 12 computer programs have been submitted earlier that day for batch processing. In how many ways can Matthew order the processing of these programs if (a) There are no restrictions? (b) He considers four of the
Patter's Pastry Parlor offers eight different kinds of pastry and six different kinds of muffins. In addition to bakery items one can purchase small, medium, or large containers of the following beverages: coffee (black, with cream, with sugar, or with cream and sugar), tea (plain, with cream, with
Calculate and check your answer by listing all the selections of size 2 that can be made from the letters a, b, c, d, e, and f.
A student is to answer seven out of 10 questions on an examination. In how many ways can he make his selection if(a) There are no restrictions?(b) He must answer the first two questions?(c) He must answer at least four of the first six questions?
In how many ways can 12 different books be distributed among four children so that(a) Each child gets three books?(b) The two oldest children get four books each and the two youngest get two books each?
How many arrangements of the letters in MISSISSIPPI have no consecutive S's?
(a) Fifteen points, no three of which are collinear, are given on a plane. How many lines do they determine?(b) Twenty-five points, no four of which are coplanar, are given in space. How many triangles do they determine? How many planes? How many tetrahedra (pyramidlike solids with four triangular
Determine the value of each of the following summations(a)(b)(c)(d)where n is an odd positive integer(e)
Express each of the following using the summation (or Sigma) notation. In parts (a), (d), and (e), n denotes a positive integer.(a)(b) 1 + 4 + 9 + 16 + 25 + 36 + 49 (c) 13 - 23 + 33 - 43 + 53 - 63 + 73 (d) (e)
For the strings of length 10 in Example 1.24, how many have (a) Four 0's, three l's, and three 2's; (b) At least eight l's; (c) Weight 4?
Consider the collection of all strings of length 10 made up from the alphabet 0, 1, 2, and 3. How many of these strings have weight 3? How many have weight 4? How many have even weight?
In the three parts of Fig. 1.8, eight points are equally spaced and marked on the circumference of a given circle.(a) For parts (a) and (b) of Fig. 1.8 we have two different (though congruent) triangles. These two triangles (distinguished by their vertices) result from two selections of size 3 from
How many triangles are determined by the vertices of a regular polygon of n sides? How many if no side of the polygon is to be a side of any triangle?
(a) In the complete expansion of (a + b + c + d) ∙ (e + f + g + h)(u + v + w + x + y + z) one obtains the sum of terms such as agw, cfx, and dgv. How many such terms appear in this complete expansion? (b) Which of the following terms do not appear in the complete expansion from part (a)? (i)
Determine the coefficient of x9y3 in the expansions of(a) (x + y)12(b) (x + 2y)12(c) (2x - 3y)12.
Complete the details in the proof of the multinomial theorem.Multinomial theorem.For positive integers n, t, the coefficient of x1n1x2n2x3n3 . . . xtnt in the expansion of (x1 + x2 + x3 + . . . + xt)n iswhere each ni is an integer with 0 ‰¤ ni ‰¤ n, for all 1 ‰¤
Determine the coefficient of (a) xyz2 in (x + y + z)4 (b) xyz2 in (w + x + y + z)4 (c) xyz2 in (2x - y - z)4 (d) xyz-2 in (x - 2y + 3z-1)4 (e) w3x2yz2 in (2w - x + 3y - 2z)8
Find the coefficient of w2x2y2z2 in the expansion of(a) (w + x + y + z + l)10(b) (2w - x + 3y + z - 2)12(c) (u + w - 2x + y + 5z + 3)12.
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) (2s - 3t + 5u + 6v - 11w; + 3x + 2y)10
For any positive integer n determine(a)(b)
Show that for all positive integers' m and n,
Evaluate each of the following.(a) C(10, 4)(b)(c) C(14, 12)(d)
For x a real number and n a positive integer, show that(a)(b)(c)
(a) If a0, a1, a2, a3 is a list of four real numbers, what is ˆ‘3i=1(a1 - a1-1)?(b) Given a list--a0, a1, a2,...,an--of n + 1 real numbers, where n is a positive integer, determine ˆ‘ni=1(c) Determine the value of
(a) Write a computer program (or develop an algorithm) that lists all selections of size 2 from the objects 1, 2, 3, 4, 5, 6. (b) Repeat part (a) for selections of size 3.
(a) How many permutations of size 3 can one produce with the letters m, r, a, f, and t? (b) List all the combinations of size 3 that result for the letters m, r, a, f, and t.
If n is a positive integer and n ‰¥ 1, prove that is a perfect square.
A committee of 12 is to be selected from 10 men and 10 women. In how many ways can the selection be carried out if(a) There are no restrictions?(b) There must be six men and six women?(c) There must be an even number of women?(d) There must be more women than men?(e) There must be at least eight
In how many ways can a gambler draw five cards from a standard deck and get(a) A flush (five cards of the same suit)?(b) Four aces?(c) Four of a kind?(d) Three aces and two jacks?(e) Three aces and a pair?(f) A full house (three of a kind and a pair)?(g) Three of a kind?(h) Two pairs?
How many bytes contain (a) Exactly two l's; (b) Exactly four l's; (c) Exactly six l's; (d) At least six l's?
In how many ways can 10 (identical) dimes be distributed among five children if(a) There are no restrictions?(b) Each child gets at least one dime?(c) The oldest child gets at least two dimes?
In how many ways can Lisa toss 100 (identical) dice so that at least three of each type of face will be showing?
Two n-digit integers (leading zeros allowed) are considered equivalent if one is a rearrangement of the other. (For example, 12033, 20331, and 01332 are considered equivalent five-digit integers.)(a) How many five-digit integers are not equivalent?(b) If the digits 1,3, and 7 can appear at most
Determine the number of integer solutions forx1 + x2 + x3 + x4 + x5 < 40,where(a) xi ≥ 0, 1 ≤ i ≤ 5(b) xi ≥ - 3, 1 ≤ i ≤ 5
In how many ways can we distribute eight identical white balls into four distinct containers so that(a) No container is left empty?(b) The fourth container has an odd number of balls in it?
(a) Find the coefficient of v2w4xz in the expansion of (3v + 2 w + x + y + z)8.(b) How many distinct terms arise in the expansion in part (a)?
In how many ways can Beth place 24 different books on four shelves so that there is at least one book on each shelf? (For any of these arrangements consider the books on each shelf to be placed one next to the other, with the first book at the left of the shelf.)
For which positive integer n will the equations(1) x1 + x2 + x3 + ... + x19 = n, and(2) y1 + y2 + y3 + ... + y64 = nhave the same number of positive integer solutions?
(a) How many nonnegative integer solutions are there to the pair of equations x1 + x2 + x3 + . . . + x7 = 37, x1 + x2 + x3 = 6?(b) How many solutions in part (a) have x1, x2, x3 > 0?
How many times is the print statement executed for the following program segment? (Here, i, j, k, and m are integer variables.)for i : = 1 to 20 dofor j : = 1 to i dofor k : = 1 to j dofor m : = 1 to k doprint (i * j) + (k * m)
In how many ways can 15 (identical) candy bars be distributed among five children so that the youngest gets only one or two of them?
In the following program segment, i, j, k, and counter are integer variables. Determine the value that the variable counter will have after the segment is executed.counter : = 10for i := 1 to 15 dofor j : = i to 15 dofor k : = j to 15 docounter : = counter + 1
Find the value of sum after the given program segment is executed. (Here i, j, k, increment, and sum are integer variables.)increment : = 0sum : = 0for i := 1 to 10 dofor j : = 1 to i dofor k : = 1 to j dobeginincrement : = increment + 1sum := sum + incrementend
Consider the following program segment, where i, j, k, n, and counter are integer variables and the value of n (a positive integer) is set prior to this segment.counter : = 0for i : = 1 to n dofor j : = 1 to i dofor k : = 1 to j docounter := counter + 1We shall determine, in two different ways, the
(a) Given positive integers m, n with m, n, show that the number of ways to distribute m identical objects into n distinct containers with no container left empty isC(m - 1, m - n) = C(m - 1, n - 1).(b) Show that the number of distributions in part (a) where each container holds at least r objects
Write a computer program (or develop an algorithm) to list the integer solutions for (a) x1 + x2 + x3 = 10, 0 ≤ xi, 1 ≤ i ≤ 3 (b) x1 + x2 + x3 + x4 = 4, - 2 ≤ xi, 1 ≤ i ≤ 4
Consider the 219 compositions of 20. (a) How many have each summand even? (b) How many have each summand a multiple of 4?
Let n, m, k be positive integers with n = mk. How many compositions of n have each summand a multiple of k?
Frannie tosses a coin 12 times and gets five heads and seven tails. In how many ways can these tosses result in(a) Two runs of heads and one run of tails;(b) Three runs;(c) Four runs;(d) Five runs;(e) Six runs;(f) Equal numbers of runs of heads and runs of tails?
(a) For n ‰¥ 4, consider the strings made up of n bits - that is, a total of n 0's and l's. In particular, consider those strings where there are (exactly) two occurrences of 01. For example, if n = 6 we want to include strings such as 010010 and 100101, but not 101111 or 010101. How many such
A certain ice cream store has 31 flavors of ice cream available. In how many ways can we order a dozen ice cream cones if (a) We do not want the same flavor more than once? (b) A flavor may be ordered as many as 12 times? (c) A flavor may be ordered no more than 11 times?
(a) In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar, and five (identical) Susan B. Anthony dollars? (b) In how many ways can we select n objects from a collection of size 2n that consists of n distinct and
Determine the number of integer solutions ofx1 + x2 + x3 + x4 -: 32,where(a) xi ≥ 0, 1 ≤ i ≤ 4(b) xi > 0, 1 ≤ i ≤ 4(c) x1, x2 ≥ 5, x3, x4 ≥ 7(d) xi ≥ 8, 1 ≤ i ≤ 4(e) xi ≥ - 2, 1 ≤ i ≤ 4(f) x1, x2, x3 > 0, 0 < x4 < 25
In how many ways can a teacher distribute eight chocolate donuts and seven jelly donuts among three student helpers if each helper wants at least one donut of each kind?
Verify that for each integer n ‰¥ 1,
(a) In how many ways can one go from (0, 0) to (7, 3) if the only moves permitted are R: (x, y) → (x + 1, y) and U: (x, y) → (x, y + 1), and the number of U's may never exceed the number of R's along the path taken?(b) Let m, n be positive integers with m > n. Answer the question posed in
Twelve patrons, six each with a $5 bill and the other six each with a $10 bill, are the first to arrive at a movie theater, where the price of admission is five dollars. In how many ways can these 12 individuals (all loners) line up so that the number with a $5 bill is never exceeded by the number
(a) In how many ways can one travel in the xy-plane from (0, 0) to (3, 3) using the moves R: (x, y) → (x + 1, y) and U: (x, y) → (x, y + 1), if the path taken may touch but never fall below the line y = xl In how many ways from (0, 0) to (4, 4)?(b) Generalize the results in part (a).(c) What
Find the other three ways to arrange 1, 2, 3, 4, 5, 6 in two rows of three so that the conditions in part (d) of Example 1.43 are satisfied.
There are b4 (= 14) ways to arrange 1, 2, 3, . . . , 8 in two rows of four so that (1) the integers increase in value as each row is read, from left to right, and (2) in any column the smaller integer is on top. Find, as in part (d) of Example 1.43, (a) The arrangements that correspond to each of
There are 132 ways in which one can parenthesize the product abcdefg. (a) Determine, as in part (c) of Example 1.43, the list of five l's and five 0's that corresponds to each of the following. (i) (((ab)c)(d(ef))) (ii) (a(b(c(d(ef))))) (iii) ((((ab)(cd))e)f) (b) Find, as in Example 1.43, the way
Consider drawing n semicircles on and above a horizontal line, with no two semicircles intersecting. In parts (a) and (b) of Fig. 1.10 we find the two ways this can be done for n = 2; the results for n = 3 are shown in parts (c)-(g).(i) How many different drawings are there for four
Mr. and Mrs. Richardson want to name their new daughter so that her initials (first, middle, and last) will be in alphabetical order with no repeated initial. How many such triples of initials can occur under these circumstances?
In how many ways can the 11 identical horses on a carousel be painted so that three are brown, three are white, and five are black?
Four numbers are selected from the following list of numbers: - 5, - 4, - 3, - 2, -1, 1, 2, 3, 4.(a) In how many ways can the selections be made so that the product of the four numbers is positive and(i) The numbers are distinct?(ii) Each number may be selected as many as four times?(iii) Each
Waterbury Hall, a university residence hall for men, is operated under the supervision of Mr. Kelly. The residence has three floors, each of which is divided into four sections. This coming fall Mr. Kelly will have 12 resident assistants (one for each of the 12 sections). Among these 12 assistants
(a) How many of the 9000 four-digit integers 1000, 1001, 1002,..., 9998, 9999 have four distinct digits that are either increasing (as in 1347 and 6789) or decreasing (as in 6421 and 8653)?(b) How many of the 9000 four-digit integers 1000, 1001, 1002...., 9998, 9999 have four digits that are either
(a) Find the coefficient of x2yz2 in the expansion of [(x/2) + y - 3z]5.(b) How many distinct terms are there in the complete expansion of(c) What is the sum of all coefficients in the complete expansion?
(a) In how many ways can 10 people, denoted A, B, I, J, be seated about the rectangular table shown in Fig. 1.11, where Figs. 1.11 (a) and 1.11(b) are considered the same but are considered different from Fig. 1.11 (c)? (b) In how many of the arrangements of part (a) are A and B seated on longer
(a) Determine the number of nonnegative integer solutions to the pair of equationsx1 + x2 + x3 = 6. x1 + x2 + ... + x5 = 15.xi ≥ 0, 1 ≤ i ≤ 5.(b) Answer part (a) with the pair of equations replaced by the pair of inequalitiesx1 + x2 + x3 ≤ 6. x1 + x2 + ... + x5 = 15.xi ≥ 0, 1 ≤ i ≤ 5.
For any given set in a tennis tournament, opponent A can beat opponent B in seven different ways. (At 6-6 they play a tie breaker.) The first opponent to win three sets wins the tournament.(a) In how many ways can scores be recorded with A winning in five sets?(b) In how many ways can scores be
Given n distinct objects, determine in how many ways r of these objects can be arranged in a circle, where arrangements are considered the same if one can be obtained from the other by rotation.
For every positive integer n, show that
Francesca has 20 different books but the shelf in her dormitory residence will hold only 12 of them.(a) In how many ways can Francesca line up 12 of these books on her bookshelf?(b) How many of the arrangements in part (a) include Francesca's three books on tennis?
Determine the value of the integer variable counter after execution of the following program segment. (Here i, j, k, l, m, and n are integer variables. The variables r, s, and t are also integer variables; their values - where r ≥ 1, s ≥ 5, and t ≥ 7 - have been set prior to this
(a) Find the number of ways to write 17 as a sum of l'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.
(a) In how many ways can 17 be written as a sum of 2's and 3's if the order of the summands is(i) Not relevant?(ii) Relevant?(b) Answer part (a) for 18 in place of 17.
(a) If n and r are positive integers with n ≥ r, how many solutions are there tox1 + x2 + ... + xr = nwhere each xt is a positive integer, for 1 ≤ i ≤ r?(b) In how many ways can a positive integer n be written as a sum of r positive integer summands (1 ≤ r ≤ n) if the order of the
(a) In how many ways can one travel in the xy-plane from (1, 2) to (5, 9) if each move is one of the following types: (R): (x, y) → (x + 1, y); (U): (x, y) (x, y + 1)? (b) Answer part (a) if a third (diagonal) move (D): (x, y) → (x + 1, y + 1) is also possible.
(a) In how many ways can a particle move in the xy-plane from the origin to the point (7, 4) if the moves that are allowed are of the form:(R): (x, y) †’ (x + 1, y); (U): (x, y) †’ (x, y + 1)?(b) How many of the paths in part (a) do not use the path from (2, 2) to (3, 2) to (4,
Twelve points are placed on the circumference of a circle and all the chords connecting these points are drawn. What is the largest number of points of intersection for these chords?
Due to their outstanding academic records, Donna and Katalin are the finalists for the outstanding physics student (in their college graduating class). A committee of 14 faculty members will each select one of the candidates to be the winner and place his or her choice (checked off on a ballot)
Consider the 8 Ã— 5 grid shown in Fig. 1.13. How many different rectangles (with integer-coordinate comers) does this grid contain? [For example, there is a rectangle (square) with comers (1, 1), (2, 1), (2, 2), (1, 2), a second rectangle with corners (3, 2), (4, 2), (4, 4), (3, 4), and a third
As head of quality control, Silvia examined 15 motors, one at a time, and found six defective (D) motors and nine in good (G) working condition. If she listed each finding (of D or G) after examining each individual motor, in how many ways could Silvia's list start with a run of three G's and have
In order to graduate on schedule, Hunter must take (and pass) four mathematics electives during his final six quarters. If he may select these electives from a list of 12 (that are offered every quarter) and he does not want to take more than one of these electives in any given quarter, in how many
In how many ways can a family of four (mother, father, and two children) be seated at a round table, with eight other people, so that the parents are seated next to each other and there is one child on a side of each parent? (Two seating's are considered the same if one can be rotated to look like
A choir director must select six hymns for a Sunday church service. She has three hymn books, each containing 25 hymns (there are 75 different hymns in all). In how many ways can she select the hymns if she wishes to select(a) Two hymns from each book?(b) At least one hymn from each book?
How many ways are there to place 25 different flags on 10 numbered flagpoles if the order of the flags on a flagpole is(a) Not relevant?(b) Relevant?(c) Relevant and every flagpole flies at least one flag?
A penny is tossed 60 times yielding 45 heads and 15 tails. In how many ways could this have happened so that there were no consecutive tails?
In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels?

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