A student can choose 1 language and 1 science from 5 languages and 4 sciences in 5 x 4 = 20 ways. Six biology books, 5 chemistry books, and 2 physics books can be arranged on a shelf in 3! x 6! x 5! x 2! = 1036800 ways so that the books of each subject stand together. The number of 5-letter words that can be formed from the letters in "chromate" is 5280 if each letter is used once, and 37,763 if letters can be repeated. A 4-digit number can be formed in 120 ways from 4 of 5 digits without repetition, and 10,000 ways with repetition. Of the numbers without repetition, 24

1. MATH 4-COMBINATORICS
and
A student has a choice of 5 foreign languages and 4 sciences. In how many ways can he choose 1 language
1
science?
a shelf so
six different biotogy books, 5 different chemistry books and 2 different physics books are to be arranged on
that the biology books stand together, the chemistry books stand togeiher and the physics books stand together'
How many such arrangements are possible?
word chromate
Determine the number of different words of 5 letters each that can be formed with the letters in the
(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.)
O" formed by using 4 out of the 5 digits 1,2,3,4,5 (a) if the digits must not be repeated
in
How many numbers r"V
nrrb"r, (b) if they'may be repeitedZ lt tne digits must not be repeated, how many of the 4-digit numbers
"ny
(c) begin with 2, (d) end with 25?
5 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?
of these
(a) How many 5-digit numbers can be formed from the '10 digits 0,1,2,9,...,9, repetitions allowed? How many
numbers (b) begin with 40, (c) are even, (d) are divisible by 5?
(a) may not be given to the
7. In how many ways can 2 different prizes be awarded among 10 contestants if both prizes
same person, (b) may be given to the same person?
d. In how many ways can 5 letters be mailed if there are 3 mailboxes available?
many ways can
o There are four iandidates for president of a club, 6 for vice-president and 2 for secretary. In how
these three positions be filled?
10. Twelve different pictures are available, of which 4 are to be hung in a row. In how many
ways can this be done?
places. How many sucn
11. It is required to seat 5 men and 4 women in a row so that the women occupy the even
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
42 together,
(b) 3 of the books are never all 3 together?
each other?
14. In how many ways can 10 women bi seated in a row so that 2 particular women will not be next to
'15. 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?
to. 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?
at a time?
1g. (a) How many arrangements can be made from ihe ietters of tne woiO cooperator when all are taken
How many of luch ariangements (b) have the three o's together, (c) begin with the two r's?
shelf?
There are3 copies eacfr-of 4 different books. In how many different ways can they be arranged on a
.19.
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?

2. MATH A_COM BINATORICS A.N SWTR KAY
1. 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) = 29
2. 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 many
such 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) il
each 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 any
number, sPq = 120 (b) if they may be repeated? 5" = $25 lf the digits must not be repeated, how many of the 4-digit
numbers (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") = 90000
How 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 same
person, 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" = 243
9. 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) = 4E
10. Twelve different pictures are available, of which 4 are to be hung in a row. In how many ways can this be done? nPa=11884
11. 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) = 2880
12. 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!) = 1440
13. 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! = 332640
14. 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! = 2903040
15. 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) = ?40
16. 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) * 23760
17. 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 be
seated 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 = 20160
22. ln each case, find n: (a) nCn-z = 10, n=5 (b) = nCr r, n.2$ (c) nPa =30 nCs. il=8
23. Given nPr = 3024 and nC, = 126, find r. r=4 "Crs
24. How many different sets of 4 students can be chosen out of 17 qualified students to represent a school in a mathematics
contest? rzCa = 2380
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? rzCe * 220
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. oC: = 20
27. How many diagonals does an octagon have? aCz * 8 = 2S
28. How many parallelograms are formed by a set of 4 parallel lines intersecting another set of 7 parallel lines? (qCz)(;C:) = 126
29. 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 - 40
30. 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 arc
always 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 so
as 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 this
be 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 consonants
and 2 different vowels? (gC:)(.aC:) . 336
34. ln how many ways can a person choose 1 or more of 4 electrical appliances? 2" * 1 = 15
35. In how many ways can 2 or more ties be selected out of 8 ties? 2" * {eCc + eC } - 247