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MATH 4-COMBINATORICSA student has a choice of 5 foreign languages and 4 sciences. In how many ways can he choose 1 language and 1

science?six different biotogy books, 5 different chemistry books and 2 different physics books are to be arranged on a shelf so

that the biology books stand together, the chemistry books stand togeiher and the physics books stand together'

How many such arrangements are possible?

Determine the number of different words of 5 letters each that can be formed with the letters in the word chromate

(a) if each letter is used not more than once, (b) if each letter may be repeated in any arrangement. (These words

need not have meaning.)How many numbers r"V O" formed by using 4 out of the 5 digits 1,2,3,4,5 (a) if the digits must not be repeated in

"ny nrrb"r, (b) if they'may be repeitedZ lt tne digits must not be repeated, how many of the 4-digit numbers

(c) begin with 2, (d) end with 25?t6* riany 4-digit numbers may be formed with the 10 digits 0,1,2,3,...,9 (a) if each digit is used only once in each

number, (b) How many of these numbers are odd?(a) How many 5-digit numbers can be formed from the '10 digits 0,1,2,9,...,9, repetitions allowed? How many of these

numbers (b) begin with 40, (c) are even, (d) are divisible by 5?

In how many ways can 2 different prizes be awarded among 10 contestants if both prizes (a) may not be given to the

same person, (b) may be given to the same person?

In how many ways can 5 letters be mailed if there are 3 mailboxes available?

There are four iandidates for president of a club, 6 for vice-president and 2 for secretary. In how many ways can

these three positions be filled?Twelve different pictures are available, of which 4 are to be hung in a row. In how many ways can this be done?

It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many sucn

arrangements are Possible?12. In how many orders can 7 different pictures be hung in a row so that 1 specified picture is (a) at the center,

(b) at either end?In how many ways can g different books be arranged on a shelf so that (a) 3 of the books are always together,

(b) 3 of the books are never all 3 together?In how many ways can 10 women bi seated in a row so that 2 particular women will not be next to each other?

How many numbers between 3000 and 5000 can be formed by using the 7 digits 0,1,2,3,4,5,6 if each digit must not

be repeated in anY number?From 11 novels and 3 dictionaries, 4 novels and 1 dictionary are to be selected and arranged on a shelf so that the

dictionary is always in the middle. How many such arrangements are possible?

17. How many signali can be made with 5 different flags by raising them any number at a time?

1g. (a) How many arrangements can be made from ihe ietters of tne woiO cooperator when all are taken at a time?

How many of luch ariangements (b) have the three o's together, (c) begin with the two r's?.19. There are3 copies eacfr-of 4 different books. In how many different ways can they be arranged on a shelf?

20. (a) In how many ways can 5 persons be seated at a round table? (b) In how many ways can 8 persons be seated at

a round table if 2 particular persons must always sit together?

21. By stringing together 9 differently colored beads, how many different bracelets can be made?

22. ln eacniaie, find n: (a) nCn-z = 10, (b)nCrs = nC1' (c)"Po =30 nCs'

23. Given nP,= 3024 and nC, = 126, find r.

24. How many different iets of 4 students can be chosen out of 17 qualified students to represent a school in a

mathematics contest?25. ln how many ways can 12 books be divided between A and B so that one may get 9 and the other 3 books?

26. Determine the number of different triangles which can be formed by joining the six vertices of a hexagon, the vertices

of each triangle being on the hexagon.27. Haw many diagonals does an octagon have?28. How many parattelograms are form-ed by a set of 4 parallel lines intersecting another set of 7 parallel lines?

2g. There are 10 pointsln a plane. No three of the points are in a straight line, except 4 points which are all in the same

straight line. How many straight lines can be formed by joining the 10 points?

30. In how many ways can 3 women be selected out oi 15 women (a) if 1 of the women is to be included in every

selection, (bi if 2 of the women are to be excluded form every selection, (c) if 1 is always included and 2 are always

excluded?31 . An organization has 25 members, 4 of whom are doctors. In how many ways can a committee of 3 members be

selected so as to include at least 1 doctor?32. From 6 chemists and 5 biologists, a committee of 7 is to be chosen so as to include 4 chemists. ln how many ways

can this be done?33. Given g consonants and 4 vowels, how many S-letter words can be formed, each word consisting of 3 different

consonants and 2 different vowels?34. In how many ways can a person choose 1 or more of 4 electrical appliances?35. In how many ways can 2 or more ties be selected out of 8 ties?

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MATH A_COM BINATORICS A.N SWTR KAY1. A student has a choice of 5 foreign languages and 4 sciences. In how many ways can he choose 1 language and 1

science? (5)(a) = 292. Six different biology books, 5 different chemistry books and 2 different physics books are to be arranged on a shelf so that

the biology books stand together, the chemistry books stand together and the physics books stand together. How manysuch arrangements are possible? (3!X6!X5!X2!) = 1036800

3. Determine the number of different words of 5 letters each that can be formed wlth the letters in the word chromate (a) ileach letter is used not more than once, aPr = S720 (b) if each letter may be repeated in any arrangement. 8' = 3?763(These words need not have meaning.)

4. How many numbers may be formed by using 4 out.of the 5 digits 1,2,3,4,5 (a) if the digits must not be repeated in anynumber, sPq = 120 (b) if they may be repeated? 5" = $25 lf the digits must not be repeated, how many of the 4-digitnumbers (c) begin with 2, aPr = 24 (d) end with 25? 3P2 = $

5. How many 4-digit numbers may be formed with the 10 digits 0,1 ,2,3,...,9 (a) if each digit is used only once in each number,(gPrXsPg) . 4536 (b) How many of these numbers are odd? (sPr)(gPrXaPz) = 2520

6. (a) How many S-digit numbers can be formed from the'10 digits 0,1,2,3,...,9, repetitions allowed? GprX10") = 90000How many of these numbers (b) begin with 40, 10'- 1000 (c) are even, (oPr)(10')(rP') = 45000 (d) are divisible by 5?(sPrX10"XzPr) * 18000

7. In how many ways can 2 different prizes be awarded among '10 contestants if both prizes (a) may not be given to the sameperson, roPz = 90 (b) may be given to the same person? 10'- = 100

8. In how many ways can 5 letters be mailed if there are 3 mailboxes available? 3" = 2439. There are four candidates for president of a club, 6 for vice-president and 2 for secretary. ln how many ways can these

three positions be filled? {4)i6XZ) = 4E10. Twelve different pictures are available, of which 4 are to be hung in a row. In how many ways can this be done? nPa=1188411. lt is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such

arrangements are possible? (5lX4l) = 288012. In how many orders can 7 different pictures be hung in a row so that 1 specified picture is (a) at the center, 6! * 7?0 (b) at

either end? (rPrX6!) = 144013. ln how many ways can 9 different books be arranged on a shelf so that (a) 3 of the books are always together,

(6+1)!31 = 30240 (b) 3 of the books are never all3 together? 9! * (6*1)13! = 33264014. In how many ways can 10 women be seated in a row so that 2 particular women will not be next to each other?

10!- (8+1)!2! = 290304015. How many numbers between 3000 and 5000 can be formed by using the 7 digits 0,1 ,2,3,4,5,6 if each digit must not be

repeated in any number? (2P1)(6P3) = ?4016. From 1'l novels and 3 dictionaries, 4 novels and 1 dictionary are to be selected and arranged on a shelf so that the

dictionary is always in the middle. How many such arrangements are possible? (rPa){rCr) * 2376017. How many signals can be made with 5 different flags by raising them any number at a time? Medyo vaguel Best answer; 25 * 1=31

18. (a) How many arrangements can be made from the letters of the word cooperator when all are taken at a time?101(312!) * 302400 How many of such arrangements (b) have the three o's together, 8!12! = 20160 (c) begin with the two/s? 8!i3l = 6V2*

19. There are 3 copies each of 4 different books. In how many different ways can they be arranged on a shelf?12!l(3!3!3!3!) = 369600

20. (a) In how many ways can 5 persons be seated at a round table? (5 - 1)l = 2a (b) In how many ways can 8 persons beseated at a round table if 2 particular persons must always sit tbgether? {7 * 1)l2l = 1440

21. By stringing together 9 differently colored beads, how many different bracelets can be made? (9 * 1)!/2 = 2016022. ln each case, find n: (a) nCn-z = 10, n=5 (b)

"Crs = nCr r, n.2$ (c) nPa =30 nCs. il=823. Given nPr = 3024 and nC, = 126, find r. r=424. How many different sets of 4 students can be chosen out of 17 qualified students to represent a school in a mathematics

contest? rzCa = 238025. ln how many ways can 12 books be divided between A and B so that one may get 9 and the other 3 books? rzCe * 22026. Determine the number of different triangles which can be formed by joining the six vertices of a hexagon, the vertices of

each triangle being on the hexagon. oC: = 2027. How many diagonals does an octagon have? aCz * 8 = 2S28. How many parallelograms are formed by a set of 4 parallel lines intersecting another set of 7 parallel lines? (qCz)(;C:) = 12629. There are 10 points in a plane. No three of the points are in a straight line, except 4 points which are all in the same straight

line. How many straight lines can be formed by joining the 10 points? :cCz - aC: + 1 - 4030. In how many ways can 3 women be selected out of 15 women (a) if 1 of the women is to be included in every selection,

r<Cz - 91 (b) if 2 of the women are to be excluded from every selection, rsC: = 28S (c) if 1 is always included and 2 arcalways excluded? r:Cz = 66

31. An organization has 25 members, 4 of whom are doctors. ln how many ways can a committee of 3 members be selected soas to include at least 1 doctor? :sCg *:rC: = $70

32. From 6 chemists and 5 biologists, a committee of 7 is to be chosen so as to include 4 chemists. In how many ways can thisbe done? (5Ca)(5C3) * 150

33. Given 8 consonants and 4 vowels, how many 5-letter words can be formed, each word consisting of 3 different consonantsand 2 different vowels? (gC:)(.aC:) . 336

34. ln how many ways can a person choose 1 or more of 4 electrical appliances? 2" * 1 = 1535. In how many ways can 2 or more ties be selected out of 8 ties? 2" * {eCc + eC } - 247