Сборник задач по высшей математике 2 том
.pdf5.5.42. |
lloKa3aTb, qTO eCJIH BeKTopHoe nOJIe F |
= f(r)·r, r,D;e r = xi+yj+zk |
|
H r = Irl, COJIeHOM,D;aJIbHO, TO f(r) = |
~. |
5.5.43. |
|
r |
BY,D;eT JIM npOCTpaHCTBeHHoe nOJIe F=r·(cxr), r,D;e r=xi+yj+zk, |
||
|
r=lrl Me - nOCTOjlHHblii BeKTOp, COJIeHOM,D;aJIbHbIM? |
|
5.5.44. |
lloKa3aTb, qTO npocTpaHcTBeHHoe nOJIe F = f(r) . r, r,D;e r(x, y, z) |
|
|
M r = Irl, nOTeHQHaJIbHO H HaitTM ero nOTeHQMaJI. |
|
5.5.45. |
lloKa3aTb, qTO eCJIH BeKTopHoe nOJIe F nOTeHQMaJIbHO, TO BeKTOp- |
|
|
HOe nOJIe c x F (r,D;e c - nOCTOjlHHblii BeKTop) jlBJIjleTCjI COJIeHO- |
|
|
M,D;aJIbHbIM. BepHo JIM 06paTHoe? |
|
5.5.46. |
BepHo JIM, qTO BeKTopHoe npOM3Be,D;eHMe nOTeHQMaJIbHbIX nOJIeii |
|
|
nOTeHQMaJIbHO? |
|
5.5.47. |
BepHo JIM, qTO BeKTopHoe npOM3Be,D;eHHe COJIeHOM,D;aJIbHbIX nOJIeii |
|
|
COJIeHOM,D;aJIbHO? |
|
5.5.48. |
lloKa3aTb, qTO BeKTopHoe npOH3Be,D;eHHe nOTeHQMaJIbHbIX nOJIeii- |
|
|
COJIeHOH,D;aJIbHOe BeKTopHoe nOJIe. |
|
5.5.49. BepHo JIH, qTO BeKTopHoe npOH3Be,D;eHHe COJIeHOH,D;aJIbHbIX noJIeit - nOTeHQMaJIbHOe BeKTopHoe nOJIe?
....
rnaBa 6. TEOPlll~ BEPO~THOCTElII
o
§1. 3flEMEHTbIKOM6~HATOP~K~
Bo MHOrJ.!X 3ap;a'laXKJIaCCH'leCKotiTeopHH BepOHTHocTeti HCnOJIb3yeTcH '1wM6u-
'ltamopU'lCa, T. e. pa3p;eJI MaTeMaTHKH, B KOTOPOM H3Y'lalOTCHpa3JIH'lHb!ecoep;HHeHHH (KOM6HHaD;HH) 9JIeMeHTOB KOHe'lHb!XMHOiKeCTB.
MHorHe KOM6HHaTOpHb!e 3ap;a'lHMoryT 6b!Tb pe!lleHb! C nOMOn:t:blO P;ByX npaBHJI - npaBHJIa YMHOiKeHHH H npaBHJIa CJIOiKeHHH.
TeopeMa 6.1. npasVlflo YMHO)KeHVl1I: eC11111 illS HeKOToporo KOHe'lHOrOMHO>KeCTBa nepBbl1il 06beKT (311eMeHT a) MO>KHO Bbl6paTb nl cnoc06aMIII, a BTOpolil 06beKT (311eMeHT b) - n2 cnoc06aMIII, TO 06a 06beKTa (a III b) B YKasaHHOM nOpl'lAKe MO>KHO Bbl6paTb nl . n2 cnoc06aMIli.
9TOT npHHD;Hn pacnpOCTpaHHeTCH Ha cJIY'latiTpex H 60JIee 06'beKToB.
TeopeMa 6.2. |
npasVlflo CflO)KeHVl1I: |
eC11111 HeKoTopbl1il 06beKT a |
MO>KHO Bbl6paTb |
|||
nl cnoc06aM III , |
a |
06beKT b MO>KHO |
Bbl6paTb n2 |
cnoc06aMIII, |
nplII'leM nepBble III |
|
BTopble cnoc06bl |
He nepeceKalOTCl'I, TO 11I0601il |
illS 06beKTOB |
(a |
11111111 b) MO>KHO |
||
Bbl6paTb nl + n2 cnoc06aMIli. |
|
|
|
|
||
9TO npaBHJIO pacnpOCTpaHHeTCH Ha JI1060e KOHe'lHOe'lHCJIO06'beKToB. Cyn:t:ecTBYlOT p;Be cxeMbI BbI60pa m 9JIeMeHTOB H3 3ap;aHHoro MHOiKeCTBa: 6e3
6036pa~e'ltUJ!, Korp;a BbI6paHHble 9JIeMeHTbI He B03Bpan:t:alOTCH B Hcxop;Hoe MHOiKeCTBO, H C 6036pa~e'ltUeM, Korp;a BbI60p ocyn:t:eCTBJIHeTCH n09JIeMeHTHO C 06H3aTeJIbHbIM B03Bpan:t:eHHeM OTo6paHHOrO 9JIeMeHTa Ha KaiKp;OM !llare.
CxeMa Bbl60pa 6e3 B03Bpaw.eHMi.1
IIycTb p;aHO MHOiKeCTBO, COCTOHn:t:ee H3 n pa3JIH'lHbIX9JIeMeHTOB.
~ Pa3Me~e'ltUeM U3 n 3.1teMe'ltm06 no k 3.1teMe'ltm06 (0 :::; k :::; n) Ha3bIBaeTCH
JIio6oe ynopHp;O'leHHOe nop;MHOiKeCTBO p;aHHoro MHOiKeCTBa, cop;epiKan:t:ee k 9JIeMeHTOB.
)J;Ba pa3Men:t:eHHH pa3JIH'lHbI,eCJIH OHH OTJIH'lalOTCHp;pyr OT p;pyra JIH60 coCTaBOM 9JIeMeHTOB, JIH60 nopHp;KOM HX paCnOJIOiKeHHH.
l.JHCJIO pa3Men:t:eHHti H3 n 9JIeMeHTOB no k
'IHCJIHeTCHno <p0pMYJIe
k |
l)(n - |
2) ... (n - |
n! |
(1.1) |
An = n(n - |
k + 1) = (n _ k)!' |
rp;e n! = 1 . 2 . 3 ... n, npH'leM1! = 1, O! = l.
271
~IIepecmanoa'K:o{}, U3 n 3J1eMenmOa Ha3bIBaeTCS'Ipa3Mem;eHHe H3 n 3JIeMeHTOB no
n 3JIeMeHTOB. |
~ |
TromM 06Pa30M, YKa3aTb Ty HJIH HHyIO nepecTaHoBKY ,lJ;aHHOrO MHOlKeCTBa H3 n 3JIeMeHTOB 3HaqHT Bbl6paTb onpe,lJ;eJIeHHblfi nOpS'l,lJ;OK3THX 3JIeMeHTOB. IIo3TOMY JII06ble ,lJ;Be nepecTaHoBKH OTJIHqaIOTCS'I,lJ;pyr OT ,lJ;pyra TOJIbKO nOpS'l,lJ;KOMCJ1e,lJ;OBaHHS'I3JIeMeHTOB.
T.JHCJ10 nepecTaHoBOK H3 n 3JIeMeHTOB 0603HaqaeTCS'ICHMBOJIOM Pn H BblqHCJ1S'1-
eTCS'Ino <popMYJIe
Pn = A~ = n!
~Co'temanueM U3 n 3J1eMenmOa no k (0 ~ k ~ n) Ha3blBaeTCS'IJII060e nO,lJ;MHo-
lKeCTBO ,lJ;aHHOrO MHolKecTBa, KOTopoe CO,lJ;eplKHT k 3JIeMeHTOB. |
~ |
JII06ble ,lJ;Ba COqeTaHHS'IOTJIHqaIOTCS'I,lJ;pyr OT ,!I;pyra XOTS'I6bl O,lJ;HHM ::meMeHTOM (T. e. OTJIHqaIOTCS'ITOJIbKO COCTaBOM 3JIeMeHTOB). T.JHCJ10 COqeTaHHfi H3 n 3JIeMeHTOB no k 0603HaqaeTCS'ICHMBOJIOM e~ H BblQHCJ1S'1eTCS'Ino <popMYJIe
ek |
_ |
n(n - 1)(n - 2) ... (n - |
k + 1) _ |
n! |
_ |
A~ |
n |
- |
k! |
- |
k!(n - |
k)! - |
k!· |
,1J;JIS'I QHCeJI e~ (OHH Ha3blBaIOTCS'I 6UnOMUaJlbn'WMu 'K:03ifiifJU?J,ueHmaMu)
JIHBbI CJ1e)J;yIOm;He TOlK,lJ;eCTBa:
(1.3)
cnpaBe,lJ;-
e~ = e;::-k (npaauJlo cU.M.Mempuu),
e~ + e~ + ... + e;:: = 2n ,
e~ = e~-l + e~=~ (npaaUJIo IIac'K:a.tIJI),
e~ = e;:: = 1.
CxeMa Bbl60pa C B03Bpaw.eH ....eM
ECJ1H npH ynOpS'l)J;OQeHHOfiBbl60pKe k 3JIeMeHTOB H3 n 3JIeMeHTbl B03Bpam;aIOT-
CS'I06paTHO, TO nOJIYQeHHble Bbl60pKH npe,lJ;CTaBJIS'lIOTco6ofi pa3MetqenWi c noamo-
peHWlMU. T.JHCJ10 BCex pa3Mem;eHHfi C nOBTOpeHHS'lMHH3 n 3JIeMeHTOB no k o603Ha-
-k
QaeTCS'ICHMBOJIOM An H BblQHCJ1S'1eTCS'Ino <p0pMYJIe
-k k |
. |
(1.4) |
An =n |
ECJ1H npH Bbl60pKe k 3JIeMeHTOB H3 n 3JIeMeHTbl B03Bpaw;aIOTCS'I06paTHO 6e3 nOCJ1e)J;yIOm;ero ynopS'l,lJ;OQHBaHHS'I(TaKHM 06Pa30M, O,lJ;HH H Te lKe 3JIeMeHTbl MoryT BblHHMaTbCS'Ino HeCKOJIbKY pa3, T. e. nOBTOpS'lTbCS'I),TO nOJIYQeHHble Bbl60pKH eCTb co""emaHWI c noamopenWI.Mu. T.JHCJ10 Bcex COQeTaHHfi C nOBTOpeHHS'lMHH3 n
3JIeMeHTOB no k 0603HaQaeTCS'ICHMBOJIOM en-k H BblQHCJ1S'1eTCS'Ino <popMYJIe
(1.5)
IIycTb B MHOlKeCTBe H3 n 3JIeMeHTOB eCTb k pa3JIHQHblX THnOB 3JIeMeHTOB, npH
3TOM THn 3JIeMeHTOB nOBTOpS'leTCS'Inl pa3, n2 pa3, ••• , nk Pa3,
272
npH'leMnl +n2+ ... +nk = n. Tor,Ll;a nepecTaHoBKH 3JIeMeHTOB ,LI;aHHOro MHOlKeCTBa npe,Ll;CTaBJIHIOT co6oil: nepeCma?W67CU C n06mOpeXtIJIMU.
lIHCJIO nepeCTaHOBOK C nOBTOpeHHHMH (HHOr,Ll;a rOBOpHT 0 'lHCJIepa36HeHHiI:
MHOlKeCTBa) H3 n 3JIeMeHTOB 0603Ha'laeTCHCHMBOJIOM Pn(nl, n2, ... , nk) H BhI'lHCJIHeTCH no <popMYJIe
|
|
(1.6) |
|
|
Ta6.11u'4a 1 |
(I-H cTpoKa - |
6e3 nOBTopeHHiI:, 2-H cTpoKa - |
C nOBTopeHHHMH) |
Pa3Mern;eHHH |
IIepecTaHoBKH |
CO'leTaHHH |
1 |
Ak _ |
n! |
||
n-(n_k)! |
||||
|
||||
2 |
-k |
|
k |
|
An =n |
|
|||
Pn =n! |
|
|
k |
n! |
|
|
Cn = k!(n - k)! |
||
|
|
|
||
Pn(nl,n2, ... ,nk)= " |
n! |
, |
-k |
k |
|
Cn = Cn+k- 1 |
|||
nl.·n2.· ... ·nk. |
|
|
||
(nl +n2+ ... +nk =n)
6.1.1.CKOJIbKO Pa3JIHqHbIX TpeX3HaqHbIX qHCeJI MO)KHO COCTaBHTb H3
|
IJ;HcPP 0, |
2, |
3, |
5, |
7 |
ec.rrH: |
|
a |
a) IJ;HcPPbl He rroBTopHIOTCHj |
6) IJ;HcPPbI MoryT rroBTopHTbCH? |
|||||
a) IIepBYIO IJ;HcPPy MO)KHO BbI6paTb qeTblpbMH crroc06aMH (qHCJIa BH.n;a |
|||||||
025, 073, ... He CqHTaeM TpeX3HaqHbIMH). Bbl6paB rrepByIO IJ;HcPPy (HarrpHMep, IJ;HcPPy 5), BTOPYIO IJ;HcPPY MO)KHO TaK)Ke BbI6paTb qeTblpbMH crroc06aMH (BTOpoii IJ;HcPpoii MO)KeT 6b1Tb JII06as H3 OCTaBIIIHXCH 0, 2, 3, 7). TpeTbIO
IJ;HcPPY, OqeBH.n;HO, MO)KHO BbI6paTb TpeMH crroc06aMH. CJIe.n;oBaTeJIbHO, comacHo rrpaBHJIY YMHO)KeHHH HMeeTCH 4 . 4 . 3 = 48 crroco6oB paccTaHoBKH IJ;HcPP, T. e. HCKOMblX TpeX3HaqHbIX qHCeJI 6y.n;eT 48 (BOT HeKOTOpbIe H3 HHX: 509, 237, 530, 702, ... ).
6) IIoHHTHo, qTO eCJIH IJ;HcPPbI MorYT rrOBTopHTbCH, TO TpeX3HaqHble qHCJIa MO)KHO COCTaBHTb 4 . 5 . 5 = 100 crroco6aMH (BOT HeKOTopble H3 HHX: 222,
200, 332, ... ). •
6.1.2.CKOJIbKO qHCeJI, co.n;ep)KaIIJ;HX He MeHee Tpex rrorrapHO Pa3JIHqHbIX
IJ;HcPP, MO)KHO COCTaBHTb H3 IJ;HcPP 2, 4, 6, 8, 9?
a IIo rrpaBHJIY YMHO)KeHHH TpeX3HaQHbIX QHCeJI MO)KHO COCTaBHTb 5 . 4 . 3 =
=60 crroco6aMH, a QeTblpeX3HaQHbIX - 5·4·3·2= 120 crroco6aMH, CTOJIbKO
)Ke rrHTH3HaQHblX QHCeJI (5·4·3·2·1)IIo. rrpaBHJIY CJIO)KeHHH, BcerO MO)KHO
COCTaBHTb 60 + 120 + 120 = 300 QHCeJI, COCTOHIIJ;HX He MeHee QeM H3 Tpex
rrorrapHO pa3JIHQHbIX IJ;HcPp. •
6.1.3.CKOJIbKHMH crroco6aMH MorYT 6b1Tb pacrrpe.n;eJIeHbI TpH rrpH30BblX MeCTa Cpe.n;H 16 copeBHYIOIIJ;HXCH?
273
6.1.4. B CTY)J;eHqeCKoil: rpynne 12 )J;eBYIIIeK U 16 IOHoIIIeil:. CKOJIbKUMU cnoc06aMH MO)KHO BbI6paTb )J;JIfI BpyqeHHfI npH30B )J;ByX CTY)J;eHTOB O)J;Horo nOJIa?
6.1.5. ECJIH nO)J;6pocHTb O)J;HOBpeMeHHO TpH HrpaJIbHbIe KOCTH, TO CKOJIb-
KO HMeeTCfI pa3JIHqHbIX B03MO)KHbIX KOM6HHaU:Hil: BbI6pOIIIeHHbIX
OqKOB?
6.1.6. B U:BeTOqHOM KHOCKe 7 BH)J;OB U:BeTOB. CKOJIbKHMH pa3HbIMH cnoc06aMH MO)KHO COCTaBHTb 6YKeT, cO)J;ep)KaIll;Hil: 3 U:BeTKa?
6.1.7.lh nyHKTa A B nyHKT B MO)KHO )J;06paTbcfI CaMOJIeTOM, noe3)J;OM,
|
aBT06ycOM, a H3 Hero B nyHKT C - |
neIIIKOM, Ha TpaKTope, Ha JIO- |
|
IIIa)J;H, Ha JIO)J;Ke. CKOJIbKHMH cnoc06aMH MO)KHO BbI6paTb )J;opory |
|
|
OT nyHKTa A )J;O nyHKTa C qepe3 B? |
|
6.1.8. |
CKOJIbKHMH cnoc06aMH MO)KHO BbI6paTb O)J;HH U:BeTOK H3 KOP3HHbI, |
|
|
B KOTOPOil: HMeeTCfI 12 rB03)J;HK, 15 p03 H 7 xpH3aHTeM? |
|
6.1.9. |
CocTaBHTb pa3JIHqHbIe pa3MeIll;eHHfI no )J;Ba 9JIeMeHTa H3 9JIeMeH- |
|
|
TOB MHO)KeCTBa A = {3, 4, 5} H nO)J;CqHTaTb HX '1HCJIO. |
|
o 113 Tpex 9JIeMeHTOB MO)KHO 06pa30BaTb CJIe)J;YIOIlI;He pa3MeIll;eHHfI no )J;Ba |
||
9JIeMeHTa: (3,4); (4,3); (3,5); (5,3); (4,5); |
(5,4). TaKHM 06pa30M, Bcero |
|
ux 6. O)J;HaKo qHCJIO pa3MeIll;eHHil: MO)KHO nO)J;CqHTaTb H no <popMYJIe (1.1):
2 |
2 |
3' |
6 |
• |
A3 = 3·2= |
6 HJIH A3 = |
(3 -·2)! = |
1=6. |
6.1.10. CKOJIbKHMH cnoc06aMH 3 Harpa)J;bI (3a I, II, III MecTa) MoryT 6bITb pacnpe)J;eJIeHbI Me)K)J;y 10 yqaCTHHKaMH copeBHoBaHHil:?
o BY)J;eM CqHTaTb, 'ITOKa)K)J;blil: yqaCTHHK copeBHoBaHHil: MO)KeT nOJIyqHTb He 60JIee O)J;HOil: Harpa)J;bI. BbI6paTb 3-x yqaCTHHKOB copeBHoBaHHil: H3 10 MO)KHO
A3 |
1O! |
= 7!·8 . 9 ·10 = 720 |
|
10 = |
(10 - |
3)! |
7! |
cnoc06aMH, TaK KaK «npH30BbIe |
TPOil:KH» |
OTJIUqalOTCfI )J;pyr OT )J;pyra JIH60 |
|
COCTaBOM yqaCTHHKOB, JIH60 nopfl)J;KOM HX CJIe)J;OBaHHfI. |
|||
8TOT )Ke pe3YJIbTaT MO)KHO nOJIyqHTb, npHMeHfIfI npaBHJIO YMHO)KeHHfI: |
|||
npeTeH)J;eHTOB Ha rJIaBHYIO Harpa)J;y (3a I |
MecTo) 9; Ha BTOPYIO - 8; Ha Tpe- |
||
TblO - 7; qUCJIO pa3JIUqHbIX cnoc060B pacnpe)J;eJIeHUfI Harpa)J; paBHO 10·9·8=
= 720. |
• |
6.1.11. |
CKOJIbKO HMeeTCfI nflTH3HaqHbIX qHCeJI, Bce U:H<PPbI y KOTOPbIX |
|
pa3JIHqHbI? |
6.1.12. CKOJIbKHMU cnoc06aMH MO)KHO COCTaBUTb TpexU:BeTHbIil: nOJIocaTbIil: <pJIar (TpU ropH30HTaJIbHbIX nOJIOCbI), ecJIH HMeeTCfI MaTepufI
5pa3JIHqHbIX U:BeToB?
6.1.13.113 rpynnbi B 15 qenOBeK BbI6upalOT 4-x yqaCTHHKoB 9cTa<peTbI
800x 400 x 200 x 100. CKOJIbKHMH cnoc06aMu MO)KHO paccTaBHTb cnopTcMeHoB Ha 9TUX 9Tanax?
6.1.14.CocTaBHTb pa3JIUqHbIe nepecTaHoBKH H3 9JIeMeHTOB MHO)KeCTBa
A={5;8;9}.
274
a ITo cP0pMYJIe (1.2) qllCJIO lIepeCTaHOBOK ll3 3-x 9JIeMeHTOB paBHO P3 =
= 3! = 1·2·3 = 6. COCTaBJI~eM llX: (5,8,9); (5,9,8); (8,9,5); (8,5,9); (9,5,8);
(9,8,5). |
4t |
6.1.15. |
CKOJIbKllMll cIIoc06aMll MO)KHO paccTaBllTb Ha KHM:)KHOit 1I0JIKe )J,e- |
|
C~TllTOMHllK IIpoll3Be)J,eHllit ,I1;. JIoH)J,oHa, paCIIOJIara~ llX: |
a) B IIPOll3BOJIbHOM 1I0p~)J,Ke;
6) TaK, qT06bI I, V II IX TOMa CTO~JIll p~)J,OM (B JII060M 1I0p~)J,Ke); B) TaK, qT06bI I, II, III TOMa He CTO~JIll p~)J,OM (B JII060M 1I0p~)J,Ke).
a a) qllCJIO CIIoc060B pacCTaHOBKll 10 KHllr paBHO qllCJIY lIepeCTaHOBOK ll3
10 9JIeMeHTOB: PlO = 1O! = 3628800.
6) |
MbICJIeHHO CB~3aB I, V II IX TOMa llJIll 1I0JIO)KllB B O)J,llH lIaKeT, IIOJIY- |
qllM 8 |
«KHllr», T. e. 7 KHllr II 1 CB~3KY (llJIll lIaKeT) KHllr. I1x MO)KHO paCCTa- |
BllTb Ha 1I0JIKe Ps = 8! clIoc06aMll. Ka)K)J,OMY ll3 9TllX cIIoc060B paCCTaHOBKll cooTBeTcTBYIOT P3 = 3! cIIoc060B paCCTaHOBKll KHllr, HaXO)J,~IIIllXC~ B CB~3Ke (I, V II IX TOMa 1I0-lIpe)KHeMY CTO~T p~)J,OM, HO B llHOM 1I0p~)J,Ke). CornacHO IIpaBllJIY yMHO)KeHll~, qllCJIO B03MO)KHbIX paccTaHoBoK 10 KHllr Ha 1I0JIKe TaK, qT06bI 3' OlIpe)J,eJIeHHble KHllrll (I, V II IX TOMa) CTO~JIll p~)J,OM, paBHo
Ps . P3 = 8! . 3! = 40320 . 6 = 241920.
B) I1cKoMoe qllCJIO CIIoc060B paCCTaHOBKll KHllr, C yqeTOM IIYHKTOB a) II
6), paBHO PlO - Ps . P3 = 3628800 - 241 920 = 3386880. |
4t |
6.1.16.B KOMHaTe llMeeTC~ 7 CTYJIbeB. CKOJIbKllMll cIIoc06aMll MO)KHO pa3-
|
MeCTllTb Ha HllX 7 rocTeit? 3 rocT~? |
|
|
||||
6.1.17. |
CTY)J,eHTbI C)J,aIOT 5 9K3aMeHOB, B TOM qllCJIe 2 9K3aMeHa IIO MaTe- |
||||||
|
MaTllKe. CKOJIbKllMll cIIoc06aMll MO)KHO paClIpe)J,eJIllTb 9K3aMeHbI, |
||||||
|
HO TaK, qT06bI 9K3aMeHbI IIO MaTeMaTllKe CJIe)J,OBaJIll O)J,HH 3a )J,py- |
||||||
|
ruM? He CJIe)J,OBaJIll O)J,llH 3a )J,PyruM? |
|
|||||
6.1.18. |
CKOJIbKO pa3JIllqHbIX «CJIOB» |
MO)KHO IIOJIyqllTb, lIepeCTaBJI~~ 6YK- |
|||||
|
BbI B CJIOBe: |
|
|
|
6) TEATP; |
|
|
|
a) COJIHIJ;E; |
|
|
|
|
||
|
B) JII1JII1; |
|
|
|
r) |
SOS? |
|
6.1.19. |
CKOJIbKllMll cIIoc06aMll MO)KHO yIIOp~)J,OqllTb MHO)KeCTBO |
A = |
|||||
|
= {8, 9,10,11, ... , 15} |
TaK, |
qT06bI Ka)K)J,oe qeTHOe qllCJIO |
llMeJIO |
|||
|
qeTHbIit HOMep? |
|
|
|
|
|
|
6.1.20. CocTaBllTb pa3JIllqHbIe |
COqeTaHll~ IIO )J,Ba ll3 9JIeMeHTOB MHO)Ke- |
||||||
|
CTBa A = {3, 4, 5} II 1I0)J,CqllTaTb llX qllCJIO. |
|
|||||
a 113 Tpex 9JIeMeHTOB MO)KHO COCTaBHTb CJIe)J.yIOIIIlle Tpll COqeTaHll~ IIO )J,Ba |
|||||||
9JIeMeHTa: {3, 4}; {3, 5}; |
{4, 5}. I1x |
qllCJIO MO)KHO 1I0)J,CqllTaTb II IIO cPop- |
|||||
MyJIe (1.3): C§ = ~:~ |
= 3 (llJIll |
TaK: C§ |
= 2!(}~ 2)! = 3; llJIll TaK: |
||||
C§ = C~ = 3). |
|
|
|
|
|
4t |
|
6.1.21. BJIa)J,llMllp XOqeT IIp"rJIaC"Tb B rOCTH TPOllX ll3 ceMll CBOllX JIyqIIIllX )J,pY3eit. CKOJIbKllMll cIIoc06aMll OH MO)KeT BbI6paTb IIpllrJIaIIIeHHbIx?
275
<;:) TaK KaK ,1l;JUI BJIa,1l;IiMlipa BaJKeH TOJIbKO COCTaB rocTeil: (rrOpH,1l;OK POJIIi
He IirpaeT), TO '1IiCJIOcrroco60B BbI60pa TPOIiX rocTeil: 1i3 7 MO:>KHO Hail:TIi rro
A,. |
( |
) |
C3 |
7 . 6 . 5 |
35. |
• |
,±,opMyJIe CO'leTaHIiil: |
1.3: |
7 = |
~ = |
|||
6.1.22. B |
Ba3e CTOHT 9 KpaCHbIX Ii 7 P030BbIX rB03,1l;IiK. CKOJIbKIiMIi crroco- |
|||||
6aMii MO:>KHO BbI6paTb 1i3 Hee: |
|
|
||||
a) |
3 rB03,1l;IiKIi; |
|
|
|
|
|
6) 6 rB03,1l;IiK O,1l;HOrO IIBeTa; |
|
|
||||
B) 4 KpaCHbIX Ii 3 P030BbIX rB03,1l;IiKIi? |
|
|||||
Q a) TaK KaK rrOpH,1l;OK BbI60pa IIBeTOB He IiMeeT 3Ha'leHIiH,TO BbI6paTb 3 rB03,1l;IiKIi 1i3 Ba3bI, B KOTOPOil: CTOHT 16 rB03,1l;IiK, MO:>KHO Cf6 crroco6aMIi. ITo
A,. |
() |
C3 |
= |
16· 15· 14 |
560. |
|
,±,opMyJIe |
1.3 HaXO,1l;IiM: |
16 |
1. 2 . 3 |
= |
||
6) BbI6paTb 6 rB03,1l;IiK KpacHoro IIBeTa MO:>KHO cg = 84 crroco6aMIi, a 6 rB03,1l;IiK p030Boro IIBeTa C¥ = 7 crroco6aMIi. ITo rrpaBIiJIy CJIO:>KeHIiH BbI6paTb
6 rB03,1l;IiK O,1l;HOrO IIBeTa (KpacHblx IiJIIi P030BbIX) MO:>KHO cg +C¥ = 84 + 7 =
=91 crroco6oM.
B)BbI6paTb 4 KpaCHbIX rB03,1l;IiKIi 1i3 9 IiMeIOIIIIiXCH MO:>KHO C~ crroco6aMIi, a 3 P030BbIX 1i3 IiMeIOIIIIiXCH 7 MO:>KHO C? crroco6aMIi. IT09TOMY 6YKeT 1i3 4
KpaCHbIX |
Ii 3 P030BbIX rB03,1l;IiK MO:>KHO COCTaBIiTb rro rrpaBIiJIy YMHO:>KeHIiH |
|||||||
4 |
C3 _ |
9! |
7! _ |
5! . 6 . 7 . 8 . 9 |
4!· 5 . 6 . 7 _ |
• |
||
Cg • |
7 |
- |
4! . 5! . 3! ·4! - |
5!· 1 . 2 . 3 . 4 . 4! . 1 . 2 . 3 - 4410 crroco6aMIi. |
||||
6.1.23. |
CKOJIbKIiMIi crroco6aMii MO:>KHO pa36liTb 8 rrpe,1l;MeTOB Ha ,1l;Be paB- |
|||||||
|
|
|
Hble (rro KOJIIi'leCTBYrrpe.JJ:MeToB) rpyrrrrbI? |
|
||||
6.1.24. |
|
rpyrrna IIIaXMaTIiCTOB CblrpaJIa Me:>K,1l;y co6oil: 28 rrapTIiil:. KaJK,1l;ble |
||||||
|
|
|
,1l;Ba 1i3 HIiX BCTpe'laJIliCbMe:>K,1l;y co6oil: O,1l;IiH pa3. CKOJIbKO IIIax- |
|||||
|
|
|
MaTIiCTOB Y'lacTBOBaJIOB copeBHoBaHIiIi? |
|
||||
6.1.25. |
|
rpyrrna TypliCTOB 1i3 12 IOHoIIIeil: Ii 7 ,1l;eBYIIIeK Bbl6lipaeT rro :>Kpe- |
||||||
|
|
|
6liIO 5 '1eJIOBeK,1l;JIH npliroTOBJIeHIiH y:>KIiHa. CKOJIbKO cYIIIecTByeT |
|||||
|
|
|
crroco6oB rrpli KOTOPbIX B 9Ty «rrHTepKy» norra,1l;yT: |
|||||
|
|
|
a) |
O,1l;HIi ,1l;eByIIIKIi; |
6) 3 IOHOIIIIi Ii |
2 ,1l;eByIIIKIi; |
||
|
|
|
B) |
1 IOHOIIIa Ii |
4 ,1l;eByIIIKIi; |
r) 5 IOHoIIIeil:? |
|
|
6.1.26. |
CKOJIbKIiMIi crroco6aMii MO:>KHO pa36liTb 9 rrpe,1l;MeTOB Ha 2 rpyrrrrbl |
|||||||
|
|
|
(BbI6op O,1l;HOil: rpyrrrrbI O,1l;H03Ha'lHOOrrpe,1l;eJIHeT BTOPYIO)? |
|||||
6.1.27. |
|
ITHTb aBTopOB ,1l;OJI:>KHbI HarrlicaTb 3a,1l;a'lHIiKrro MaTeMaTIiKe, COCTO- |
||||||
|
|
|
HIIIIiil: 1i3 14 rJIaB. ,ILBa aBTopa HarrliIIIYT rro 2 rJIaBbI, ,1l;Ba,1l;pyrlix- |
|||||
|
|
|
rro 3 Ii eIIIe O,1l;IiH - 4 rJIaBbI KHlirli. CKOJIbKIiMIi crroco6aMii MO:>KeT |
|||||
|
|
|
6bITb pacnpe,1l;eJIeH MaTepliaJI Me:>K,1l;y aBTopaMIi? |
|||||
6.1.28.B HIIIIiKe 15 ,1l;eTaJIeil:, cPe.JJ:1i KOTOPbIX 6 6paKOBaHHblx. HaY,1l;a'lY
BbI6lipaeTCH KOMnJIeKT 1i3 5 ,1l;eTaJIeil:. CKOJIbKO Bcero KOMnJIeKTOB,
B KaJK,1l;OM 1i3 KOTOPbIX 2 ,1l;eTaJIli 6paKoBaHHble?
6.1.29.113 9JIeMeHTOB (IIIi<PP) 2, 4, 5 COCTaBIiTb Bce pa3MeIIIeHliH Ii CO'le-
TaHIiH C rroBTopeHIiHMIi rro ,1l;Ba 9JIeMeHTa.
Q Pa3MeIIIeHIiH c nOBTopeHIiHMIi no ,1l;Ba 9JIeMeHTa TaKOBbI: (2,2); (2,4);
(2,5); (4,4); (4,5); (4,2); (5,5); (5,2); (5,4).
276
Mx qUCJIO MO>KHO BhlqUCJIUTh U IIO <popMYJIe (1.4):
-2 |
= 3 |
2 |
= 9. |
A3 |
|
COqeTaHUH C IIOBTOpeHUHMU IIO ,n:Ba 9JIeMeHTa TaKOBhI (B OTJIHqUe OT pa3MenreHUfi 3,n:eCh IIOPH,n:OK 9JIeMeHTOB B BhI60pKe He UMeeT 3HaqeHHH, T. e., HaIIpUMep, IIapbI (2,4) U (4,2) He pa3JIUqaIOTCH): {2,2}; {2,4}; {2,5}; {4,4};
{4,5}; {5,5}.
Mx qUCJIO MO)l{HO BhlqUCJIUTb U IIO <popMYJIe (1.5):
|
-2 |
2 |
2 |
4·3 |
= 6. |
• |
|
C 3 |
= C3+2-1 |
= C 4 |
= D |
||
6.1.30. B Mara3UHe UMeeTCH 7 BU,n:OB TOpTOB. CKOJIhKUMU cIIoc06aMU MO)l{- |
||||||
|
HO COCTaBUTh Ha60p, co,n:ep)l{anrufi 3 TopTa? A |
eCJIU UMeIOTCH 3 |
||||
a |
Bu,n:a TOPTOB, a HY)I{eH Ha60p H3 7 TOpTOB? |
|
||||
TIOCKOJIbKY IIOPH,n:OK paCIIOJIO)l{eHUH TOPTOB B Ha60pe He urpaeT pOJIU, |
||||||
TO UCKOMoe qUCJIo. Ha60pOB paBHO qUCJIY COqeTaHUfi C IIOBTopeHUHMU U3 7 9.neMeHToB IIO 3 B K~,n:OM. TIo <popMYJIe (1.5) UMeeM C~ = C~ = i:~:~ = 84
(CM. TaK)I{e 3a,n:aqy 6.1.6). |
|
ECJIU UMeeTCH 3 Bu,n:a TOPTOB, a HY)I{eH Ha60p U3 7 TOPTOB, TO qUCJIO |
|
B03MO)l{HhIX Ha60pOB paBHO C~ = CJ = i:~ = 36. |
• |
6.1.31. TIHTh qeJIOBeK BOWJIU B JIU<pT Ha 1-M 9T~e ,n:eBHTU9T~Horo,n:o Ma. CKOJIbKHMU cIIoc06aMu IIacc~uphI MorYT BblfiTU U3 JIu<pTa Ha HY)I{HhIX 9T~ax?
a K~,n:hlfi U3 5 IIacc~upOB MO)l{eT BblfiTU Ha JII060M U3 BOChMU 9T~efi co
2-ro IIO 9-fi BKJIIOqHTeJIbHO. B03MO)l{HbIMU BapuaHTaMU ux Bhlxo,n:a HBJIHIOTCH, HaIIpUMep, 2-3-5-5-5 (9TO 3HaqUT, qTO Ha 2-M 9T~e BbIWeJI O,n:UH IIacc~up,
Ha 3-M - |
O,n:UH, a Tpoe BhIWJIU Ha 5-M 9T~e) UJIU 9-9-9-9-9 UJIU 4-5-6-7-9, |
|||||
U T.,n:. |
|
|
|
|
|
|
06nree qUCJIO BhIXO,n:OB IIacc~upOB, IIO <popMYJIe (1.4), paBHo |
||||||
|
|
-5 |
5 |
= 32768. |
|
|
|
|
As |
= 8 |
|
||
8TOT )l{e pe3YJIbTaT MO)l{HO IIOJIyqUTb, UCIIOJIh3YH IIpaBUJIO YMHO)l{eHuH: |
||||||
,n:JIH 1-ro IIacc~Hpa UMeeTCH 8 |
BapuaHTOB Bblxo,n:a Ha 9T~e, ,n:JIH 2-ro - |
|||||
TO)l{e 8, U ,n:JIH 3-ro - |
8, U ,n:JIH 4-1'0- |
|
8, U ,n:JIH 5-ro - 8. Bcero IIOJIyqaeTCH |
|||
8 . 8 . 8 . 8 . 8 = 8 5 BapuaHToB Bblxo,n:a 5-TU IIacc~upoB. |
• |
|||||
6.1.32. |
CKOJIhKO pa3JIUqHbIX «CJIOB» |
(IIO,n: «CJIOBOM» |
IIOHUMaeTCH JII06aH |
|||
|
KOM6uHan:uH 6YKB) MO)l{HO COCTaBUTh, IIepeCTaBJIHH 6YKBbI B CJIOBe |
|||||
|
ArA? MISSISSIPPI? |
|
|
|
|
|
a Bo06nre U3 Tpex 6YKB MO)l{HO COCTaBHTb P3 = 3! = 6 pa3JIHqHhIX Tpex6y- |
||||||
KBeHHhlX «CJIOB». B |
CJIOBe ArA 6YKBa A IIOBTopHeTCH, a IIepeCTaHOBKa O,n:H- |
|||||
HaKOBbIX 6YKB He MeHHeT «CJIOBa». TI09TOMY qUCJIO IIepecTaHoBoK C IIOBTOpeHUHMH MeHhwe qHCJIa IIepeCTaHOBOK 6e3 IIOBTopeHHfi BO CTOJIbKO pa3, CKOJIbKO MO)l{HO IIepecTaBJIHTh IIOBTOpHIOnrueCH 6YKBhI. B ,n:aHHOM CJIOBe ,n:Be 6yKBbI (1-H U 3-H) IIOBTOPHIOTCH; II09TOMY Pa3JIHqHhIX Tpex6YKBeHHhIX «CJIOB»
277
P3 |
3' |
= 3. BnpOqeM, OT- |
U3 6YKB CJIOBa ArA MO)KHO COCTaBUTb CTOJIbKO: P2 |
= 2i |
BeT MO)KHO nOJIyqUTb U nporn;e: Ka)K,ll;Oe CJIOBO U3 6YKB A, r U A 0,ll;H03HaqHO
onpe,ll;eJI.HeTC.H nOJIO)KeHUeM 6YKBbI r; ux Bcero TpU, n09TOMY U Pa3JIUqHbIX CJI0B 6Y,ll;eT TO)Ke TpU.
llOJIb3Y.HCb cP0PMYJIOit (1.6), 9TOT pe3YJIbTaT MO)KHO nOJIyqUTb cpa3y:
P3 (2, 1) |
= |
3! |
3. |
|
|
|
|
2!1! = |
|
|
|
|
|||
no 9TOit )Ke cP0pMYJIe Hait,ll;eM qUCJIO 0,ll;UHHa,ll;u;aTu6YKBeHHblx «CJIOB» |
|||||||
npu nepecTaHoBKe 6YKB B CJIOBe MISSISSIPPI. 3,ll;eCb n = |
11, nl = 1, n2 = |
4 |
|||||
(4 6YKBbI S), n3 = |
4 (4 6YKBbI I), n4 = 2, n09TOMY |
|
• |
||||
|
P l1 (1,4,4,2) = |
11! |
5 ·6· 7·8 ·9· 10 . 11 |
|
|||
|
1!4!4!2! = |
1.24.2 |
=34650. |
||||
|
|
|
|
|
|
|
|
6.1.33. |
CKOJIbKUMU cnoc06aMu MO)KHO pa3MeCTUTb B ,ll;ByX KOMHaTax 9 |
||||||
|
|
Pa3JIUqHbIX npe,ll;MeToB? |
|
|
|
||
6.1.34. |
CKOJIbKUMU cnoc06aMu MO)KHO pacnpe,ll;eJIUTb 6 pa3HbIX KHur Me- |
||||||
|
|
)K.ll:Y 3 IIIKOJIbHUKaMu? |
|
|
|
||
6.1.35. |
|
B nOqTOBOM OT,ll;eJIeHUU npO,ll;aIOTC.H OTKpbITKU 6 BU,ll;OB. CKOJIbKU- |
|||||
|
|
MU cnoc06aMu MO)KHO npH06pecTu B HeM 4 OTKpbITKU? 40,ll;UHaKO- |
|||||
|
|
BbIX OTKpbITKU? 4 pa3HbIX OTKpbITKU? |
|
|
|||
6.1.36. |
CKOJIbKO pa3JIUqHbIX 6YKeToB no 5 U;BeTKOB B Ka)K,ll;OM MO)KHO co- |
||||||
|
|
CTaBUTb, eCJIU B HaJIUqUU eCTb ,ll;OCTaTOqHO MHoro U;BeTKOB qeTblpex |
|||||
|
|
BU,ll;OB? |
|
|
|
|
|
6.1.37. |
|
Y Bpaqa eCTb |
3 BU,ll;a O,ll;HOrO JIeKapCTBa, 2 BU,ll;a - ,ll;pyroro U |
4 |
|||
|
|
BMa - |
TpeTbero. B TeqeHUe ,ll;eB.HTU ,ll;Heit OH Ka)K,ll;blit ,ll;eHb npe,ll;- |
||||
|
|
JIaraeT 60JIbHOMY no O,ll;HOMY JIeKapcTBy. CKOJIbKUMU cnoc06aMu |
|||||
|
|
OH MO)KeT BbI,ll;eJIUTb 60JIbHOMY JIeKapcTBa? |
|
|
|||
6.1.38. |
CKOJIbKO cyrn;ecTByeT pa3JIUqHbIX nepecTaHoBoK 6YKB B CJIOBe |
||||||
|
|
TPAKTAT? A B «CJIOBe» |
AAAYYAYYYY? |
|
|
||
6.1.39. CKOJIbKO MO)KHO COCTaBUTb qeTblpeX3HaqHbIX qHCeJI TaK, QT06bI
JII06bIe ,ll;Be COCe,ll;HUe U;UcPPbI 6bIJIU pa3JIUQHbIMU?
6.1.40. CKOJIbKO Pa3JIUQHbIX «CJIOB», |
COCTO.Hrn;UX U3 Tpex 6YKB, MO)KHO |
06pa30BaTb U3 6YKB CJIOBa |
|
a) JIETO; |
6) IllAJIYH? |
CKOJIbKO cyrn;ecTByeT IIIeCTU3HaQHbIX QUceJI, y KOTOPbIX Ha QeTHbIX MeCTax CTO.HT QeTHbIe U;UcPPbI?
20 CTY,ll;eHTOB 06MeHUBaIOTC.H cPoToKapTOQKaMu. CKOJIbKO cPoToKapTOQeK nOHa,ll;06uJIOCb ,ll;JI.H 9Toro?
Ka)K,ll;oro U3 6 CTY,ll;eHTOB MO)KHO HanpaBUTb M.H npOXO)K,ll;eHU.H npaKTUKU Ha O,ll;HO U3 Tpex npe,ll;npU.HTuit. CKOJIbKUMU Pa3JIUQHbIMU cnoc06aMH 9TO MO)KHO ocyrn;ecTBuTb?
278
6.1.44. CKOJIbKO TpeX3HRQHbIX quceJI MO:>KHO COCTaBUTb H3 II;H<pP 0, 1, 2,
3,4 eCJIu:
a) II;U<PPbI He MoryT IIOBTOPflTbCfI;
6) II;U<PPbI MoryT IIOBTOPUTbCfI;
B)qUCJIa ,n:OJI:>KHbI 6bITb qeTHbIMU (II;U<PPbI MoryT IIOBTOpflTbCfI);
r)qUCJIO ,n:OJI:>KHO ,n:eJIUTbCfI Ha 5 (II;U<PPbI He MoryT IIOBTOpflTbCfI).
6.1.45.4 IIuaHUCTa, 5 CKpUIIaqeil: U 6 6MHUCTOB yqacTBYIOT B KOHKypce. CKOJIbKUMU cIIoco6aMu :>KIOpU MO:>KeT oTo6paTb IIO TpU IIo6e,n:uTeJIfI B Ka:>K,n:Oil: HOMuHaII;uu?
6.1.46.CKOJIbKUMU cIIoco6aMu MO:>KHO COCTaBUTb TpexII;BeTHblil: (TpU BepTUKaJIbHble IIOJIOCbI) IIOJIOCaTblil: <pJIar, eCJIU UMeeTCfI MaTepUaJI KpacHoro, :>KeJITOrO, 3eJIeHOrO U qepHOrO II;BeToB, IIpUqeM U3BeCTHO, qTO o,n:Ha U3 IIOJIOC ,n:OJI:>KHa 6bITb 3eJIeHoil:?
6.1.47.B KJIaCCe u3yqaeTCfI 7 IIpe,n:MeTOB. CKOJIbKUMU cIIoco6aMU MO:>KHO COCTaBUTb paCIIucaHue Ha IIOHe,n:eJIbHUK, eCJIH B 9TOT ,n:eHb ,n:OJI:>KHO 6bITb 5 pa3JIUqHbIX IIpe,n:MeToB?
6.1.48.CKOJIbKUMU cIIoco6aMu MO:>KHO pacCa.D:UTb BOKpyr KpyrJIOro CTOJIa 6 MaJIbqUKOB U 6 ,n:eBOqeK, eCJIU Ka:>K,n:afl ,n:eBOqKa ,n:OJI:>KHa cu,n:eTb Me:>K.n:y ,n:ByMfI MaJIbqUKaMu?
6.1.49.CKOJIbKUMU cIIoco6aMu MO:>KHO C<P0pMupoBaTb :>KeJIe3Ho,n:opO:>KHblil: COCTaB U3 9 BarOHOB TaK, qT06bI 2-il: U 4-il: BarOHbI IIIJIU qepe3 O,n:UH?
6.1.50.CKOJIbKO pa3JIUqHbIX UHUII;UaJIOB (<J)110) MO:>KHO 06pa30BaTb, ucIIOJIb3Yfl 5 IIepBbIX 6YKB pYCCKoro aJI<paBuTa?
6.1.51.113 rpyIIIIbI, CocToflIII;eil: U3 7 MY:>KqUH U 4 :>KeHIII;UH, HY:>KHO BbI6paTb 6 qeJIOBeK TaK, qTo6bI cpe,n:u HUX 6blJIO He MeHee 2-x :>KeHIII;HH. CKOJIbKUMU cIIoco6aMU 9TO MO:>KHO c,n:eJIaTb?
6.1.52.CKOJIbKUMU cIIoco6aMu MO:>KHO pacIIpe,n:eJIUTb 36 UrpaJIbHbIX KapT IIOPOBHY Me:>K.n:y qeTblpbMfI urpoKaMu?
6.1.53. |
CKOJIbKO pa3JIUqHbIX KOM6uHaII;uil: U3 6 KapT co,n:ep:>KaT 3 ,n:aMbI, |
|
2 KOPOJIfI U 1 TY3? |
6.1.54. |
113 rpyIIIIbI B 12 qeJIOBeK Ha,n:o BbI6paTb 2 qeJIOBeKa ,n:JIfI BbIIIOJI- |
|
HeHUfI O,n:HOil: pa60TbI U 3 - ,n:JIfI ,n:pyroil:. CKOJIbKUMH cIIoco6aMH |
|
9TO MO:>KHO c,n:eJIaTb? |
6.1.55. CKOJIbKO qUceJI, 60JIbIIIUX 100, MO:>KHO 3aIIucaTb C IIOMOIII;bIO II;u<PP
0, 1, 3, 5, 6 (6e3 IIoBTopeHuil:)?
6.1.56.B <pYT6oJIbHOil: KOMaH,n:e UMeeTCfI 13 IIOJIeBbIX urpOKOB U 2 Bpa-
Tapfl. CKOJIbKUMU cIIoco6aMu MO:>KHO BbI6paTb urpaIOIII;uil: COCTaB, COCTOflIII;uil: U3 10 urpOKOB U l-ro BpaTapfl?
6.1.57. CKOJIbKUMU cIIoco6aMu MO:>KHO pacIIpe,n:eJIUTb 6 6UJIeTOB B TeaTp IIO TpeM rpYIIIIaM IIepBoKYPCHUKOB?
6.1.58.B IIo,n:'be3,n:e,n:OMa YCTaHOBJIeH 3aMOK C KO,n:OM. ,il;Bepb aBTOMaTUqe-
CKU OTIIupaeTCfI, eCJIU B oIIpe,n:eJIeHHoil: IIOCJIe,n:OBaTeJIbHOCTU Ha- :>KaTb 4 KHOIIKU U3 UMeIOIII;HXCfI 12. HeKTO, He 3HM Ko,n:a, CTaJI Hay,n:aqy Ha6upaTb pa3JIUqHble KOM6uHaII;HU U3 4-x II;U<PP. KaKoe
279