This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 145 Name: §0L U T [‘0 U5 NIidterm exam 1 Student ID#
February 2 2007 INSTRUCTIONS o Read instructions carefully. 0 DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO 0 Fill in the information on this page NOW (your name, I Show your work on every problem. Correct answers with no support work will
not receive full credit. 0 N 0 EXTRA ASSISTA N CE ALLOWED. Assistance from
calulators is prohibite classmates, notes, books, 0 Use the backside of the paper too. Maximum possible Score I “I
“I
“I “I Total points 2:) h h 1. (8 points) (a) How many distinct monomials are there in the expansion of ($1 + $2 + ' ~  + 1%)"? (b) A walk starting at the origin makes either plus steps (1, 1) or minus steps (1, —1). How many
walks are there from (0,0) to (311, n) that go through (n + 2,71 — 2) but not through (2n, 0)? ® (X f*1*,+ $tg>q : (Y‘¢*xl*'(¥k)(x,+¥L) mi‘xkh)... (X‘+XL;__1¥k) {ems (A "L‘ “(WSW "'12 04 He (MM XT'XSI... XEK 1v( )hfmfw)1 “49+ «Wilt“Um: f) _. (\Jok “44 qv’ﬁletﬁ‘fn
(”Mk toMW‘lhon 0‘ A MR It Pad—S (Son/' o( Hz ’1‘;'} («46“ 0) “Us 4h 5 J Jlstm‘c) mono/“leis Au (”MLJ)
tcI 1L! smug «3 4k #0! V015 (m JUf/rébk +0 k (Ltlo‘fﬁa l) FLA/HZ; ' W5 ﬂuid “xvi(
w t (we) 4» gm) L cm) t as w w. J or Mu» w M t (M) i, (912) W" a"
1 1 , C L
«F V‘llu, va‘ , (0:01) ‘9 (Ml‘n'l) ‘ (‘1’?) $5 51% Lula (>
h h (At’zlﬂ'l)~7 URI/x) .' (22:; T; (0..) d 6* W‘L‘ 414‘! Jpn {, byleI/‘qk (:4 “N! 6 Hill“)
I. ., (nu2,4504 (141°) ’ ‘ “9”. {L [rent “4." fl” kﬁKLJ 1;“)qu e) ‘73 (0)0) 1‘ \a I1A..\——)(1I\.I\\ ;  1'9 IIGI W110 «(ram (Mm) 1‘5 (P1L) I) 1Q 57% 1s IK
I‘VMQV o£ WWII” (Ab/1K (010) I11 (P'M1OL’A) .(1 points) Lat 11 > 1. How many sulllenus (11111111111 11 11111110 111) from 11mm 1 11111 B do not 101111111
2 H! s in succession? Hint: ﬁnd a 1"‘0c1111ence. (1:3
n:\ ”=7. ‘
'AB ABA
2% 3533 ”5 5
{5 8/1 SIM; '
316/1 [AH 670M ’1": d: 0(367W1Lc5 1° 6“ (Cy/1R4 swat 5510 4 1 since “1 “WM (an01 («11mg In AX 1/1 1
MW A ﬁF/ers 1.1 M»$‘ IMMJBP—(W (Q gamﬁD/I (3.1 a, 8'
tv m ' HDMV‘F WLA q 6 OCH/’5 [1qu ({ @IIDWLA (>11 MY {61? (1} L195 ’10
(ﬁat/1? on It; r01 6( Hz £57134“). Thu, (oAswU‘ Vibﬁbcs o'€ [Wyn/1 (1, Z (5503 4%. (ex 1 salvo/M1 SI‘IIS \~I 6% ’1 I (€912: SGML: SIﬁ/I’S RIG” A_ Haw \v H‘Wt ABR 11 04 {wk «21qu is G(,\7_>‘ ‘4 _ W ”:1 111.5, \M 11M (71,1): (201,1) “1011) 111111 ., (“dram/1 ‘1 ‘IL' Féohqcu' ((71:44 Ezl F231 F5" 2 E35 E15 \
47—2 (1123 (2,5,, ‘1st 3. (8 points) (a) How many permutations of {1,2, . . . ,3k} have no multiple of 3 as a ﬁxed point? (b) Suppose that Alice has 3 nephews and ten parcels. In how many ways can she distribute the parcels if —the parcels are indistinguishable? the parcels are distinguishable? It is not required that each child receives a parcel. ® Ugo. ‘lle. IAelLstoA 651’lele fVlNlPLQ I \ 3) { ’ l /
. . 2 Lllfa‘l’ \Bu’, 'f M \n 7'
v 1 0A .. MthlAS w M fﬂ/ (
W [lb’kl Elf/V1 l?!“ 5 W lmll .4 3624 6k /‘
l4 . l4 ,1 ‘
: g ('4) (I. > (3k )
‘ IV a}; 5
{M l :0, 2k um m L Wu (2L: . ﬂ!
' [69k 94+ l° 55—4 5/
on». is 9de ‘llm 4% 2014‘ k+ 45 “Cl (#51641.
lM‘l’ 5%le lecl‘ W D im‘J. ‘1 [o fmm‘u l“ 3
® 9 (”fl ill (0 1) la ’ 3 (”(5 tang/:1”; L. “1.41% {alt/m) 5 C '. clmler,
QM ts jzwj 4. (4 points) Chocolate eggs contain stickers which can be blue, red or yellow. Each egg contains
exactly one sticker. What is the minimal number of eggs Alice needs to buy if she wants to be sure
to get either 4 blue stickers or 3 red stickers or 2 yellow stickers? Justify your answer. "( {51H
3 M 1mm. 7k W 99) qu‘ (2‘0) l ” yeam’V, quokw ll? (”ribs/”e . {Lt kl“ l/V‘Hg 0M (I {Q dew/{J ””40"“ b, Krill/mm, [4 (an Q 1212/. '{nyl ll’HJ [‘3 ﬂ“ QVYAI‘ # o( €33! 5(1 L193 A ‘u7 . {ali‘q ‘Qw {(Qhe (,1
WWIH (4' ‘84 or {(e o’f}//T'o/ fuhllAe—J no ...
View
Full Document
 Spring '07
 Peche
 Combinatorics

Click to edit the document details