Coelhos e Lotarias - Gazeta de matemática

Propaganda
```Coelhos e Lotarias
ANT&Oacute;NIO PEREIRA ROSA
ESCOLA SECUND&Aacute;RIA MARIA AM&Aacute;LIA VAZ DE CARVALHO
[email protected]
16
GAZETA DE MATEM&Aacute;TICA r174
1. INTRODU&Ccedil;&Atilde;O
lacionado com um conhecido problema de lotarias, a saber,
1XPFRQKHFLGROLYURGHSUHSDUDomRSDUDRH[DPHQDFLRQDO
TXDODSUREDELOLGDGHGHQXPFRQFXUVRGRJpQHURGR7RWROR-
GH0DWHPiWLFD\$GRDQRGHHVFRODULGDGH&gt;@SiJLQD
to com bolas numeradas de 1 a nVDtUHPGRLVQ~PHURVFRQVH-
PDU&acute;3UREOHPDGRVFRHOKRV&micro;
JHPVXJHULGDHP&gt;@HGHVHQYROYLGDHP&gt;@H&gt;@SDUHFHQRV
H[HUFtFLR VXUJH R VHJXLQWH SUREOHPD D TXH YDPRV FKD-
“Numa coelheira h&aacute; 20 coelhos sendo 5 brancos e 15 pretos.
Abre-se a porta da coelheira e deixa-se sair os coelhos um a um.
Qual a probabilidade de que saiam, pelo menos, 2 coelhos brancos
seguidos?”
FXWLYRVTXDQGRVHID]XPDH[WUDomRGHm bolas. Esta aborda-
PDLVLQWHUHVVDQWHHLQVWUXWLYDGRTXHDVLPSOHVUHVROXomRGR
SUREOHPDGRVFRHOKRVSRUPHLRGRVSURFHVVRVTXHLQGLFDPRV
QDSUy[LPDVHFomR
2. RESOLU&Ccedil;&Otilde;ES DIRETAS DO PROBLEMA
2OLYURHPFDXVDDSUHVHQWDDSHQDVDUHVSRVWDDHVWHSUR-
EOHPDVHPGDUTXDOTXHULQGLFDomRVREUHRSURFHV-
so seguido para a obter 6HWHQWDUPRVUHVROYHUHVWHSUREOHPD
1
SHORVPpWRGRVXVXDLVQRDQRFiOFXORGDSUREDELOLGDGH
GRDFRQWHFLPHQWRFRQWUiULRHDSOLFDomRGD5HJUDGH/DSODFH
YHUL&Agrave;FDPRVUDSLGDPHQWHTXHDSDUWHGLItFLOpDGHWHUPLQDomR
GRQ~PHURGHVHTXrQFLDVGHFRHOKRVEUDQFRVHSUHWRV
TXH n&atilde;oWrPGRLVFRHOKRVEUDQFRVVHJXLGRV
Este tipo de problema de contagem surge com alguma freTXrQFLDQDOLWHUDWXUDYHMDVHSRUH[HPSOR&gt;@RX&gt;@HPTXH
VHSHUJXQWDTXDORQ~PHURGHVHTXrQFLDVELQiULDV , de com2
primento n, com m]HURVHPTXHQXQFD&Agrave;JXUDPGRLV]HURV
DOS COELHOS
\$SUHVHQWDPRV HP VHJXLGD XPD VROXomR HOHPHQWDU SDUD R
SUREOHPD GRV FRHOKRV TXH HQFRQWUiPRV HP &gt;@ 3HQVDQGR
HPWHUPRVGHVHTXrQFLDVGHFRHOKRVEUDQFRV%HSUHWRV3
YDPRVFDOFXODUDSUREDELOLGDGHGRDFRQWHFLPHQWRFRQWUiULR
LVWR p D SUREDELOLGDGH GH QXQFD VXUJLUHP GRLV % VHJXLGRV
3DUDTXHQmRVXUMDPGRLV%FRQVHFXWLYRVWHPGHKDYHUSHOR
PHQRV XP 3 D VHSDUiORV FRPR DFRQWHFH SRU H[HPSOR QD
VHTXrQFLDVHJXLQWH
&laquo;%3%33%3%3&laquo;
0DLVSUHFLVDPHQWHDVHJXLUDFDGDXPGRVTXDWURSULPHLURV
FRHOKRV % WHUi GH VDLU REULJDWRULDPHQWH XP FRHOKR 3 SHOR
FRQVHFXWLYRVRXDLQGD&gt;@SiJLQD
TXH RV FRHOKRV EUDQFRV DSHQDV SRGHP RFXSDU SRVLo}HV
EOHPD H PRVWUDUHPRV WDPEpP TXH HOH HVWi GLUHWDPHQWH UH-
YDGR7HRUHPD3RULVVRHVWHVFRHOKRVSRGHPVHUGLVSRV-
6HJXLGDPHQWHYDPRVDSUHVHQWDUVROXo}HVSDUDHVWHSUR-
XPDPRWLYDomRSDUDHVWHIDFWRpGDGDPDLVjIUHQWHQDSUR-
tos de 16 A5 PDQHLUDVGLVWLQWDV\$OpPGLVVRFRPRRVFRHOKRV
SUHWRVSRGHPSHUPXWDUHQWUHVLGHPDQHLUDVFRQFOXtPRV
TXHRQ~PHURGHFDVRVIDYRUiYHLVD&acute;QmRVDtUHPGRLVFRHOKRV
EUDQFRVFRQVHFXWLYRV&micro;p 16 A5 &times; 15!'DGRTXHRVFRHOKRV
SRGHPSHUPXWDUHQWUHVLGHPDQHLUDVGLVWLQWDVDSURED-
bilidade de n&atilde;o VDtUHP GRLV FRHOKRV EUDQFRV VHJXLGRV p GH
U
m problema de c&aacute;lculo
FRPELQDWyULRHSUREDELOLGDGHV
QRDQROHYDQRVjVORWDULDV
DRVQ~PHURVGH)LERQDFFLH
FODURDRWULkQJXORGH3DVFDO
16 A
5
&times; 15!/20! = 91/323 H D UHVSRVWD DR SUREOHPD LQLFLDO p
1 − 91/323 = 232/323.
2EVHUYHVHTXHVHGLYLGLUPRVDPERVRVWHUPRVGDIUDomR
16 A
5
&times; 15!/20! por 15! &times; 5!, a probabilidade pode ser escrita
na forma 16 C5 /20 C5TXHVXJHUHXPDIyUPXODJHUDOSDUDHVWH
WLSRGHTXHVW}HV2PHVPRVXFHGHFRPRSUREOHPDGHFRQ-
1
No mesmo livro (p&aacute;gina 178) surge um outro problema semelhante a este,
referente &agrave; disposi&ccedil;&atilde;o de livros numa prateleira de uma estante, apresentando-se de novo apenas a solu&ccedil;&atilde;o 11/13.
2
Sequ&ecirc;ncias de zeros e uns.
COELHOS E LOTARIAS rAnt&oacute;nio Pereira Rosa
17
WDJHPUHIHULGRQDVHFomRVHKRXYHUm]HURVQXPDVHTXrQ-
cia bin&aacute;ria de comprimento n D FRQVLGHUDomR GD H[SUHVVmR
n−m+1 C /n C EHP FRPR R IDFWR EHP FRQKHFLGR GH KDYHU
m
m
n C VXFHVV}HVELQiULDVGHVWHWLSRVXJHUHTXHKiH[DWDPHQWH
m
n−m+1 C VHTXrQFLDV VHP ]HURV FRQVHFXWLYRV 9DPRV DJRUD
m
SURYDUDFRUUHomRGHVVDH[SUHVVmR
Teorema 1: H&aacute;
n−m+1 C VXFHVV}HVELQiULDVGHFRPSULPHQm
3HQVDQGRQR7RWRORWRSRUWXJXrVpLPHGLDWRTXHRQ~PH-
UR GH FKDYHV SRVVtYHLV p 49 C6 = 13983816 UHVWD VDEHU TXDO
RQ~PHURGHFDVRVIDYRUiYHLVDRDFRQWHFLPHQWR&acute;VDLUFKDYH
VHPQ~PHURVFRQVHFXWLYRV&micro;&amp;RQVLGHUHPRVXPDFKDYHRUGHQDGDFRPHVVDSURSULHGDGHSRUH[HPSOR
HWLUHPRVUHVSHWLYDPHQWHHDFDGDXPGRVVHXV
WHUPRV2EWHPRVDQRYDVHTXrQFLD
H
to n, compreendendo m zeros (e, logo, n − m uns) sem zeros
FRQVHFXWLYRV
2VVHXVWHUPRVHVWmRFHUWDPHQWHHQWUHHMiTXHVXE-
Demonstra&ccedil;&atilde;o: &amp;RPHFHPRVSRUHVFUHYHURVn − mVtPERORV
WUDtPRVQ~PHURVHQWUHHDQ~PHURVHQWUHHSRURXWUR
SDoRVHQWUHRVVtPERORV&acute;&micro;LQFOXLQGRRHVSDoRDQWHVGRSUL-
YRVQDVHJXQGDRVWHUPRVVmRWRGRVGLVWLQWRV6HUHSDUDUPRV
HYDPRVHVFROKHUmGHOHVSDUDFRORFDURVVtPERORV&acute;&micro;RTXH
GH VHLV Q~PHURV QDWXUDLV HQWUH H H OKH VRPDUPRV GD
&acute;&micro;&laquo;'HSRLVFRORTXHPRVRVmVtPERORV&acute;&micro;QRVHV-
PHLUR&acute;&micro;HRHVSDoRGHSRLVGR~OWLPR+in − m + 1HVSDoRV
SRGHVHUIHLWRH[DWDPHQWHGH n−m+1 Cm
maneiras, concluindo
(VWiDVVLPUHVROYLGRRSUREOHPDGRVFRHOKRVDSUREDELOL-
GDGHGHQmRVDtUHPGRLVFRHOKRVEUDQFRVVHJXLGRVp
=16
TXHRSURFHVVRpUHYHUVtYHOLVWRpTXHGDGDXPDVHTXrQFLD
PDQHLUDGHVFULWDRVQ~PHURVHREWHPRVFHUWDPHQWH XPD VHTXrQFLD GH VHLV Q~PHURV QDWXUDLV QmR FRQVH-
VHDVVLPDGHPRQVWUDomR
20−5+1 C /20 C
5
5
ODGRFRPRQDSULPHLUDVHTXrQFLDQmRKiQ~PHURVFRQVHFXWL-
C5 /20 C5 .
3UREOHPDV GHVWH JpQHUR IRUDP DERUGDGRV SHOD SULPHLUD
FXWLYRV H HQWUH H FRQFOXtPRV TXH H[LVWH XPD ELMHFomR
entre o conjunto A GDV VHTXrQFLDV RUGHQDGDV GH VHLV Q~-
PHURV QDWXUDLV HQWUH H H R FRQMXQWR B GDV VHTXrQFLDV
RUGHQDGDV GH VHLV Q~PHURV QDWXUDLV QmR FRQVHFXWLYRV HQ-
WUH H \$VVLP Card( A) = Card( B) H FRPR REYLDPHQWH
YH]SRU\$EUDKDPGH0RLYUH3QRVpFXOR;9,,,YHMDVH&gt;@QR
Card( A) =44 C6 YHPTXH Card( B) =44 C6 e a probabilidade
PRVWURXTXHDVXDVROXomRSRGHVHUREWLGDGLUHFWDPHQWHSRU
8P SRXFR PDLV IRUPDOPHQWH YDPRV SURYDU R VHJXLQWH
VpFXOR ;,; R PDWHPiWLFR LQJOrV :LOOLDP \$OOHQ :KLWZRUWK4
PHLRGRFiOFXORGRVFRH&Agrave;FLHQWHVGHFHUWRVSROLQyPLRVROHLWRU
LQWHUHVVDGRSRGHFRQVXOWDU&gt;@SiJLQDSURSRVLomR/,,
pGH 44 C6 /49 C6 ≈ 0, 5048.
teorema:
Teorema 2: 6HMDPn ≥ 1 e m ≥ 0GRLVQ~PHURVLQWHLURV6HMDP
3. O PROBLEMA DA LOTARIA
&amp;RQVLGHUHPRVXPDORWDULDHPTXHGHXPDXUQDFRQWHQGRn
bolas, numeradas de 1 a nVHH[WUDHPVHPUHSRVLomRm bolas
, e n&atilde;o h&aacute; em X n&uacute;PHURVFRQVHFXWLYRV
n = { X ⊆ {1, 2, ...n + 1 − m } : Card ( X ) = m }
Gm
.
SRUH[HPSORQRQRVVR7RWRORWRn = 49 e m = 63UHWHQGHVH
(QWmRHVWHVGRLVFRQMXQWRVWrPRPHVPRQ~PHURGHHOHPHQWRV
YRVRXPDLVJHUDOPHQWHDSUREDELOLGDGHGHDGLIHUHQoDPt-
Demonstra&ccedil;&atilde;o:%DVWDFRQVWUXLUXPDELMHFomRHQWUHRVGRLV
VDEHUTXDODSUREDELOLGDGHGHQmRVDtUHPQ~PHURVFRQVHFXWL-
QLPDHQWUHTXDLVTXHUGRLVQ~PHURVVDtGRVVHULJXDOD k &gt; 0
D QmR H[LVWrQFLD GH Q~PHURV FRQVHFXWLYRV FRUUHVSRQGH D
k = 2+iPXLWDVPDQHLUDVGHUHVROYHUHVWHSUREOHPDDUHIH-
conjuntos.
6HMD { a1 , . . . , am } ∈ Fmn SRGHPRV GHVGH Mi VXSRU TXH
ai &lt; ai+1 para cada i ∈ {1, . . . , m − 1}.3RQGRbk = ak + 1 − k ,
UrQFLD&gt;@DSUHVHQWDTXDWURSURFHVVRVGLVWLQWRV(VFROKHPRV
YHP TXH 1 ≤ b1 &lt; b2 &lt; . . . &lt; bn ≤ n + 1 − m &amp;RP HIHLWR p
sante. Tal como sucede com o problema dos coelhos, a parte
bm = am − m + 1 ≤ n − m + 1 = n + 1 − m;
DVHJXQGDGDVVROXo}HVGH&gt;@TXHpSDUWLFXODUPHQWHLQWHUHV-
GLItFLO p D GHWHUPLQDomR GR Q~PHUR GH FDVRV IDYRUiYHLV DR
DFRQWHFLPHQWR&acute;QmRVDtUHPQ~PHURVFRQVHFXWLYRV&micro;RQ~PHURGHFDVRVSRVVtYHLVpVLPSOHVPHQWH n Cm .
18
GAZETA DE MATEM&Aacute;TICA r174
yEYLRTXHb1 ≥ 1 e
TXDQWRDRUHVWRFRPRai+1 − ai ≥ 2 YHPTXH
bi + 1 − bi = a i + 1 − a i − 1 ≥ 1 e
portanto, bi+1 &gt; bi .
3RGHPRVHQWmRHVWDEHOHFHUDELMHFomRSUHWHQGLGDDVVRFLDQdo a cada { a1 , . . . , am } ∈
Fmn
(supondo os elementos de A es-
critos por ordem crescente) o conjunto B = {b1 , . . . , bm }, com
bk = ak + 1 − k GR TXH IRL GLWR DQWHULRUPHQWH UHVXOWD TXH
n 5HVWDQRVSURYDUDLQMHWLYLGDGHHDVREUHMHWLYLGDGH
B ∈ Gm
4XDQWRjLQMHWLYLGDGHVHGRLVVXEFRQMXQWRVGH Fmn , digamos
{ a1 , . . . , am } e { a1∗ , . . . , a∗m }WLYHVVHPDPHVPDLPDJHPYLULD
TXH ak + 1 − k = a∗k + 1 − k para cada k ∈ {1, . . . , m} e, portanto, ak = a∗k para todo o k ∈ {1, . . . , m}3DUDYHUL&Agrave;FDUDVRn , basta considerar
EUHMHWLYLGDGH GDGR B = {b1 , . . . , bm } ∈ Gm
os elementos a1 , . . . , am GH&Agrave;QLGRV SRU ak = bk + k − 1 para
cada k ∈ {1, . . . , m}SDUDVHFRQFOXLUTXH
UDGDVGHD6HFRQVLGHUDPRVDVSUREDELOLGDGHVDQWHULRU-
PHQWHFDOFXODGDVYHUL&Agrave;FDPRVTXHRMRJRpIDYRUiYHOjEDQFD
PDVSRUPXLWRSRXFRVHDDSRVWDIRUGHHXURDHVSHUDQoD
GHOXFURGDEDQFDp(= 1 &times; 0, 5048 − 1 &times; (1 − 0, 5048)),
OLJHLUDPHQWHLQIHULRUDXPFrQWLPRSRUFDGDHXURDSRVWDGR
\$SUHVHQWDPRV HP VHJXLGD XPD JHQHUDOL]DomR GR 7HRUH-
ma 2.
Teorema 3: O n&uacute;mero de maneiras distintas f k (m, n) de es-
colher m n&uacute;meros em {1, 2, . . . , n}WDLVTXHDPHQRUGDVGLVWkQFLDVHQWUHGRLVTXDLVTXHUGHOHVpLJXDOD k &gt; 0pGDGRSRU
f k (m, n) =n−(k−1)(m−1) Cm , com n &gt; m &gt; 1, k ≥ 1
n
B = {b1 , . . . , bm } ∈ Gm
pDLPDJHPGH { a1 , . . . , am } ∈ Fmn SRUPHLRGDDSOLFDomRFRQ-
R7HRUHPDpSUHFLVDPHQWHRFDVRk = 2).
Demonstra&ccedil;&atilde;o:1mRYDPRVDSUHVHQWiODROHLWRULQWHUHVVD-

Corol&aacute;rio:
Card ( Fmn )
=
GRSRGHFRQVXOWDU&gt;@RX&gt;@RQGHVXUJHPYiULDVGHPRQVWUD-
o}HVTXHVmRGHXPPRGRJHUDOVHPHOKDQWHVjVGR7HRUHPD
n − m +1 C .
m
HPERUDDOJRPDLV&acute;SHVDGDV&micro;

Demonstra&ccedil;&atilde;o: decorre imediatamente do teorema e do
facto de
n
Gm
ter precisamente
n − m +1 C
m
Corol&aacute;rio: 1XPDORWDULDHPTXHDVERODVWrPRVQ~PHURV
elementos.

5HSDUHPRVDJRUDTXHXPDVHTXrQFLDELQiULD B pode re-
SUHVHQWDUXPDVHTXrQFLDGHQ~PHURVQDWXUDLV N VHFRQYHQ-
&laquo; n H D FKDYH WHP m Q~PHURV D SUREDELOLGDGH GH TXH
SHOR PHQRV GRLV GRV Q~PHURV GD FKDYH WHQKDP GLVWkQFLD
inferior a kpGH
FLRQDPRVTXH
pk (m, n) = 1 −
D8P&acute;&micro;QDnpVLPDSRVLomRGHB VLJQL&Agrave;FDTXHRQ~PHURn
Demonstra&ccedil;&atilde;o: eLPHGLDWDDSDUWLUGR7HRUHPD
ID]SDUWHGDVHTXrQFLD N .

E8P&acute;&micro;QDnpVLPDSRVLomRGHB VLJQL&Agrave;FDTXHRQ~PHUR
n n&atilde;oID]SDUWHGDVHTXrQFLD N .
3RUH[HPSORjVHTXrQFLDGHQ~PHURVQDWXUDLVFRU-
UHVSRQGHDVHTXrQFLDELQiULDHYLFHYHUVD&amp;RP
HVWDFRUUHVSRQGrQFLDYHUL&Agrave;FDVHTXHDXPDVHTXrQFLDELQi-
ria de comprimento n FRQWHQGR H[DWDPHQWH k VtPERORV &acute;&micro;
QmRFRQVHFXWLYRVHVWiDVVRFLDGDXPDVHTXrQFLDGHk elemen-
tos de {1, 2, &middot; &middot; &middot; , n}VHPQ~PHURVFRQVHFXWLYRV
\$VVLPDVROXomRREWLGDSDUDRSUREOHPDGDORWDULDOHYD-
QRVDXPDVROXomRDOJRLQGLUHWDGRSUREOHPDGHFRQWDJHP
1RUHVWRGHVWDVHFomRYDPRVSURFHGHUDXPDDQiOLVHPDLV
&Agrave;QDGDVFKDYHVGDORWDULDQXPDGLUHomRGLIHUHQWHGR7HRUH-
PD XVDQGR LGHLDV TXH HQFRQWUiPRV HP &gt;@ &amp;RPHFHPRV
SRUXPDGH&Agrave;QLomR
'H&Agrave;QLomR&amp;RQVLGHUHPRVr n&uacute;meros distintos,
a1 , a2 , . . . , ar ∈ Xn = {1, 2, . . . , n},
ordenados por ordem crescente. Um grupo GD VHTXrQFLD
a1 , a2 , . . . , ar p
(i) um conjunto singular formado por um desses n&uacute;meros,
VXEMDFHQWHDRSUREOHPDGRVFRHOKRVRTXHSRVVLELOLWDDVROX-
TXDQGRDGLVWkQFLDDTXDOTXHUGRVRXWURVGDVHTXrQFLD
omRTXDVHLPHGLDWDGHVWH~OWLPR
O Teorema anterior fornece a base para um curioso jogo
n−(k−1)(m−1) C
m
.
nC
m
for superior a 1.
ou
GHD]DUVXSRQGRSDUD&Agrave;[DULGHLDVTXHn Hm SRGH-
PRVDSRVWDUQDVDtGDGHGRLVQ~PHURVFRQVHFXWLYRVTXDQGR
UHWLUDPRVDRDFDVRVHLVERODVGXPDXUQDFRPERODVQXPH-
3
Abraham de Moivre (1667-1754), matem&aacute;tico ingl&ecirc;s de origem francesa.
4
William Allen Whitworth (1840-1905), matem&aacute;tico e sacerdote anglicano.
COELHOS E LOTARIAS rAnt&oacute;nio Pereira Rosa
19
(ii) um subconjunto de dois ou mais destes n&uacute;meros, con-
Demonstra&ccedil;&atilde;o: &Eacute; semelhante &agrave; do Teorema 2. O n&uacute;mero
VHFXWLYRVTXDQGRDGLVWkQFLDGHFDGDXPGRVQ~PHURV
GH KLSyWHVHV SDUD D HVFROKD GH XPD VHTXrQFLD a1 , a2 , . . . , ar
GRVXEFRQMXQWRDTXDOTXHUGRVRXWURVGDVHTXrQFLDTXH
n&atilde;o do subconjunto) for superior a 1.
3RU H[HPSOR QD VHTXrQFLD WHPRV RV
JUXSRVFRPXPHOHPHQWR^`H^`RJUXSRFRPGRLVHOHPHQ-
WRV^`HRJUXSRFRPWUrVHOHPHQWRV^`
6HFKDPDUPRVgDRQ~PHURGHJUXSRVQDVHTXrQFLDd ao
H[WUDtGDGH Xn e com gJUXSRVp n−r+1 Cg , se ignorarmos as
SHUPXWDo}HVGRVJUXSRVSHORUDFLRFtQLRIHLWRQDSURYDGRUHIHULGR7HRUHPDSRURXWURODGRSDUDFDGDFRQ&Agrave;JXUDomRFRPg
g!
grupos de dWLSRVH[LVWHP g1 !&times; g2 !&times;...&times; gd ! possibilidades, pela
IyUPXOD GDV SHUPXWDo}HV FRP UHSHWLo}HV &amp;RPELQDQGR RV
GRLVUHVXOWDGRVYHPTXHN =
TXHUtDPRV
g!
g1 !&times; g2 !&times;...&times; gd !
&times;n−r+1 Cg , como
n&uacute;mero de diferentes tipos de grupos (n&uacute;meros isolados, pa-

UHVGHQ~PHURVFRQVHFXWLYRVWHUQRVGHQ~PHURVFRQVHFXWL-
3RU H[HPSOR VH Xn = {1, 2, . . . , 30} e considerarmos
YRVHWFH gi (i = 1, 2, . . . , d) ao n&uacute;mero de grupos de tipo ip
yEYLRTXH g1 + g2 + . . . + gd = g.
1RH[HPSORDQWHULRUg d H
g1 + g2 = 2 + 1 + 1 = 4 = g.
2UHVXOWDGRTXHSUHWHQGHPRVSURYDUpRVHJXLQWH
Teorema 4:6HMDXn = {1, 2, . . . , n}HFRQVLGHUHPRVDVVHTXrQ-
FKDYHV FRP VHWH HOHPHQWRV FRP GRLV JUXSRV FRP XP
elemento, um grupo com dois elementos e um grupo
FRP WUrV HOHPHQWRV YHUL&Agrave;FDPRV TXH H[LVWH XP WRWDO GH
3!
2!&times;1!&times;1!
&times;30−7+1 C3 = 6072 FKDYHVXPDGHODVp
TXHYLPRVDQWHULRUPHQWH
'HL[DPRV DR FXLGDGR GR OHLWRU D YHUL&Agrave;FDomR GD WDEHOD
TXHLOXVWUDDVLWXDomRSDUDR7RWRORWRQDFLRQDOn H
cias de r n&uacute;meros distintos a1 , a2 , . . . , ar H[WUDtGDVGHXn .'HV-
r diferentes, com gi (i = 1, 2, . . . , d) grupos de tipo ipGDGRSRU
g!
N=
&times; n −r +1 C g .
g1 ! &times; g2 ! &times; . . . &times; g d !
13983816 =49 C6 , RQ~PHURGHFKDYHVH[LVWHQWHVQR7RWRORWR
WDVVHTXrQFLDVRQ~PHURNGDVTXHWrPg grupos, de d tipos
5HSDUHVHTXHDVRPDGRVQ~PHURVGD~OWLPDFROXQDp
SRUWXJXrV
N&uacute;m. de
tipos de
grupos
(d)
de grupo (g i)
N&uacute;m.
total de
grupos
(g)
N
6HLVQ~PHURVLVRODGRV
1
g1 = 6
4XDWURQ~PHURVLVRODGRVHGRLVFRQVHFXWLYRV
2
g1 = 4, g2 = 1
7UrVQ~PHURVLVRODGRVHWUrVFRQVHFXWLYRV
2
g= 3, g2 = 1
'RLVQ~PHURVLVRODGRVHTXDWURFRQVHFXWLYRV
2
g1 = 1, g2 = 1
8PQ~PHURLVRODGRHFLQFRFRQVHFXWLYRV
2
g1 = 1, g2 = 1
2
6HLVQ~PHURVFRQVHFXWLYRV
1
g1 = 1
1
'RLVQ~PHURVLVRODGRVHGRLVSDUHVGHQ~PHURVFRQVHFXWLYRV
2
g1 = 2, g2 = 2
7UrVSDUHVGHGRLVQ~PHURVFRQVHFXWLYRV
1
g1 = 3,
8PQ~PHURLVRODGRXPSDUXPWHUQRGHQ~PHURVFRQVHFXWLYRV
4XDWURQ~PHURVFRQVHFXWLYRVXPSDUGHQ~PHURVFRQVHFXWLYRV
2
g1 = 1, g2 = 1
2
'RLVSDUHVGHWUrVQ~PHURVFRQVHFXWLYRV
1
g1 = 2
2
Tipo de Chave
Tabela 1
20
GAZETA DE MATEM&Aacute;TICA r174
g1 = 1, g2 = 1
g3 = 1
4. RESOLU&Ccedil;&Otilde;ES DIRETAS DO PROBLEMA DE
Teorema 6 (segundo lema de Kaplansky): O n&uacute;mero g(n, m)
CONTAGEM DOS NATURAIS N&Atilde;O CONSECUTIVOS
de maneiras de escolher m objetos dentre n objetos dispostos
2SUREOHPDGHFRQWDJHP&acute;'HWHUPLQDURQ~PHURGHPDQHLUDV
VREUHXPDFLUFXQIHUrQFLDVHPTXHKDMDGRLVFRQVHFXWLYRVp
de selecionar m objetos de entre n objetos numerados de 1 a n
HGLVSRVWRVQXPD&Agrave;ODVHPTXHVXUMDPREMHWRVFRPQ~PHURV
FRQVHFXWLYRV&micro;pPXLWRDQWLJRHSRGHVHUUHVROYLGRGHYiULDV
maneiras 'HQWUHDVYiULDVVROXo}HVFRQKHFLGDVHVFROKHPRV
5
XPDTXHpHPQRVVDRSLQLmRSDUWLFXODUPHQWHLQWHUHVVDQWH
(P ,UYLQJ .DSODQVN\ DSUHVHQWRX XPD UHVROXomR
6
elementar do famoso Probl&egrave;me des M&eacute;nages de &Eacute;douard Lucas
YHMDVH&gt;@&gt;@&gt;@RX&gt;@EDVHDGDHPGRLVOHPDVTXHVmR
FRQKHFLGRV QHVWH FRQWH[WR FRPR SULPHLUR H VHJXQGR OHPDV
n n−m
Cm , para
n−m
n &gt; m.
5. UMA RELA&Ccedil;&Atilde;O COM OS N&Uacute;MEROS DE FIBONACCI
2SUREOHPDGDVORWDULDVSHUPLWHREWHUXPDFXULRVDUHODomR
HQWUH RV FRH&Agrave;FLHQWHV ELQRPLDLV H RV Q~PHURV GH )LERQDFFL
QmRpGHHVSDQWDUTXHHVWHVVXUMDPHPSUREOHPDVUHODFLRQD-
GRVFRPFRHOKRV&laquo;1D&Agrave;JXUDDEDL[RHVWiUHSUHVHQWDGRRWULkQJXORGH3DVFDOEHPFRPRDVVXDV&acute;GLDJRQDLVVHFXQGiULDV&micro;
GH.DSODQVN\2SULPHLUROHPDGH.DSODQVN\pSUHFLVDPHQWH
XPDUHVROXomRGRSUREOHPDGHFRQWDJHPTXHWHPRVYLQGRD
estudar.
Teorema 5 (primeiro lema de Kaplansky): O n&uacute;mero
f (n, m) de maneiras de escolher m objetos dentre n obje-
WRV DOLQKDGRV VHP TXH KDMD GRLV FRQVHFXWLYRV p GDGR SRU
f (n, m) =n−m+1 Cm .
Demonstra&ccedil;&atilde;o: e yEYLR TXH f (n, 1) = n =n−1+1 C1 H TXH
f (n, n) = 0 =n−n+1 Cn , se n &gt; 1.
6HMDHQWmRmWDOTXH1 &lt; m &lt; n.3RGHPRVFRQVLGHUDUGRLV
WLSRVGHVHOHo}HVQDVFRQGLo}HVGRHQXQFLDGRDTXHODVDTXH
RSULPHLURREMHWRSHUWHQFHHDTXHODVDTXHHVWHQmRSHUWHQFH
Figura 1
4XDQWRjVSULPHLUDVpyEYLRTXHQmRSRGHPFRQWHURVH-
gundo objeto e s&atilde;o em n&uacute;mero de f (n − 2, m − 1)TXDQWRjV
Teorema 7: O n&uacute;mero total de maneiras de selecionar n&uacute;-
o n&uacute;mero f (n, m)YHUL&Agrave;FDDUHODomR
REWHQKDQ~PHURVFRQVHFXWLYRVLQFOXLQGRDVHOHomRYD]LDp
segundas, s&atilde;o em n&uacute;mero de f (n − 1, m).'LVWRGHFRUUHTXH
f (n, m) = f (n − 2, m − 1) + f (n − 1, m).
(1)
3RGHPRV HQWmR SURYDU D UHODomR f (n, m) =n−m+1 Cm por
LQGXomR&amp;RPRn &gt; m &gt; 1,FRPHFHPRVSRUQRWDUTXH
f (3, 2) = 1
= 3−2+1
meros dentre os n SULPHLURV Q~PHURV QDWXUDLV VHP TXH VH
igual a f n+2, o (n&sup2;pVLPRQ~PHURGH)LERQDFFL
Demonstra&ccedil;&atilde;o:\$SURYDpPXLWRVHPHOKDQWHjGR7HRUHPD
6HMD an R Q~PHUR D GHWHUPLQDU 4XDQGR HVFROKHPRV XQV
tantos elementos de {1, 2. . . . , n}VHPTXHKDMDQ~PHURVFRQ-
C2 .
3RUKLSyWHVHGHLQGXomR
VHFXWLYRVKiGXDVKLSyWHVHVPXWXDPHQWHH[FOXVLYDV
f (n − 1, m) =n−m Cm e f (n − 2, m − 1) =n−m Cm−.1 ;
1) O n&uacute;mero n est&aacute; entre os escolhidos.
VXEVWLWXLQGRHPYHPTXH
ou
f (n, m) =n−m Cm +n−m Cm−1 =n−m+1 Cm ,
2) O n&uacute;mero n n&atilde;o est&aacute; entre os escolhidos.
SRU XPD FRQKHFLGD SURSULHGDGH GDV FRPELQDo}HV FRPR
TXHUtDPRV
5

\$ WtWXOR GH FXULRVLGDGH HQXQFLDPRV R VHJXQGR OHPD GH
.DSODQVN\FXMDGHPRQVWUDomRSRGHVHUYLVWDHP&gt;@
Veja-se, por exemplo, [13]. O autor da solu&ccedil;&atilde;o a&iacute; apresentada, Thomas Muir
(1884 -1934), foi um especialista em determinantes; uma outra solu&ccedil;&atilde;o, recorrendo a uma equa&ccedil;&atilde;o diofantina, aparece em [3], p&aacute;gina 37.
6
COELHOS E LOTARIAS rAnt&oacute;nio Pereira Rosa
21
1R FDVR HVWDPRV HP SUHVHQoD GH XPD GDV an−1 se-
GH )LERQDFFL RV n&uacute;meros de Lucas7 R OHLWRU LQWHUHVVDGR SRGH
OHo}HV VHP Q~PHURV FRQVHFXWLYRV TXH VH SRGH ID]HU HP
FRQVXOWDU&gt;@SDUDDSURYDGHVWDD&Agrave;UPDomR
VHOHomRHRVUHVWDQWHVHOHPHQWRVGHVWDFRQVWLWXHPFHUWDPHQWH
de elementos de {1, 2, . . . , n − 2} VHP Q~PHURV FRQVHFXWLYRV
tUDPSDUDPHOKRUDUDTXDOLGDGHGRDUWLJR
WDQWHVQRFDVRGHDVHOHomRQRFDVRVHUHGX]LUDn'LVWRUH-
5. REFER&Ecirc;NCIAS
{1, 2, . . . , n − 1}QRFDVRn − 1 n&atilde;o faz certamente parte da
uma das an−2SRVVtYHLVPDQHLUDVGHHVFROKHUXPFHUWRQ~PHUR
5HSDUHVHTXHSRGHGDUVHRFDVRGHQmRKDYHUHOHPHQWRVUHV-
VXOWDTXHRQ~PHURGHVHOHo}HVFRQWHQGRnpSUHFLVDPHQWHan−2
e, portanto, an = an−1 + an−2&amp;RPRHVWDPRVDLQFOXLUDVHOHomR
YD]LDYHPTXHa1 = 2 e a2 = 3 e a sucess&atilde;o ( an )p&laquo;
sendo assim an = f n+2 , o (n&sup2;pVLPRQ~PHURGH)LERQDFFL

9HMDPRV XPD DSOLFDomR DR HVWXGR GR WULkQJXOR GH 3DVFDO
&amp;RQVLGHUHVHn HHQXPHUHPRVDVVHOHo}HVGHHOHPHQWRVGH
^`VHPHOHPHQWRVFRQVHFXWLYRV
6HOHomRYD]LD
WH[WRDVFRUUHo}HVHVXJHVW}HVTXHDSUHVHQWRXPXLWRFRQWULEX-
&gt;@\$30&sup2;4XHVWmRUHVSRQGLGDVREUHR&acute;SUREOHPDGRV
FRHOKRV&micro;UHFXUVR&acute;3HUJXQWD\$JRUD&micro;\$VVRFLDomRGRV3URIHVsores de Matem&aacute;tica.
&gt;@%DODNULVKQDQ9.Theory and Problems of Combinatorics0F*UDZ+LOO,QF1HZ&lt;RUN
6HOHo}HVFRPXPHOHPHQWR&sup2;VmRDVDEHU&acute;&micro;&acute;&micro;&acute;&micro;H&acute;&micro;
6HOHo}HVFRPGRLVHOHPHQWRV&sup2;VmRDVDEHU&acute;&micro;&acute;&micro;
H&acute;&micro;
+iXPWRWDOGH TXHpSUHFLVDPHQWHRQ~PH-
URGH)LERQDFFL
&gt;@&amp;KHQ&amp;+H.RK.0Principles and Techniques in
Combinatorics:RUOG6FLHQWL&Agrave;F6LQJDSXUD
&gt;@'UDNDNLV.Facta Universitatis: Mathematics and Informatics9ROXPH,VVXHSS

5HODFLRQHPRVHVWHVUHVXOWDGRVFRPDIyUPXODGRFRUROiULR
&gt;@+RQVEHUJHU5Mathematical Gems III (Dolciani Mathematical Expositions, No. 9), The Mathematical Association
do Teorema 2:
1) Neste caso, n m H n−m+1 Cm =5 C0 = 1
2) Neste caso, n m=1 e n−m+1 Cm =4 C1 = 4
1HVWHFDVRn m=2 e n−m+1 Cm =3 C2 = 3
e a igualdade acima pode ser escrita como
5C
0
+4
C1
+3
of America.
&gt;@ +RZDUG ) 7 &acute;7KH 3UREDELOLW\ RI &amp;RQVHFXWLYH
1XPEHUVLQ/RWWR\$SSOLHG3UREDELOLW\7UXVW&micro;GLVSRQtYHOHP
KWWSPVDSSOLHGSUREDELOLW\RUJGDWD&Agrave;OHV\$UWLFOHVSGI
C2 = f 6 ,
RXVHMDDVRPDGRVHOHPHQWRVGDGLDJRQDOVHFXQGiULDpSUHFLVDPHQWHLJXDODRQ~PHURGH)LERQDFFL
&gt;@.DSODQVN\,&acute;6ROXWLRQRIWKH&acute;3UREOqPHGHV0pQDges””, Bull. Amer. Math. Soc
O caso geral pode ser tratado da mesma maneira, cheganGRVHjFRQFOXVmRGHTXHDVRPDGDn&sup2;pVLPDGLDJRQDOVHFXQGi-
ULDGRWULkQJXORGH3DVFDOpLJXDODRn&sup2;pVLPRQ~PHURGH)LER-
&gt;@/LQKDUHV\$VG&acute;&amp;RHOKRVHP)XJD&micro;VLWH0DWHPiWLFDFRP
(http://matematica.com.sapo.ptVHFomR&acute;3UREDELOLGDGHV&micro;
LJXDOGDGH HQYROYHQGR RV Q~PHURV f (n, m) do primeiro lema
GH.DSODQVN\PDVSDUHFHQRVPDLVLQWHUHVVDQWHHDSHODWLYDD
YHUVmRHQYROYHQGRRWULkQJXOR3DUDFRQFOXLUUHIHULPRVTXHVH
pode fazer um estudo semelhante para os n&uacute;meros g(n, m) do
VHJXQGROHPDGH.DSODQVN\REWHQGRVHHPYH]GRVQ~PHURV
22
GAZETA DE MATEM&Aacute;TICA r174
&gt;@0DWWVRQ-U+)Discrete Mathematics with applications-RKQ:LOH\DQG6RQV1HZ&lt;RUN
&gt;@0F3KHUVRQ,H+RGVRQ'&acute;/RWWHU\&amp;RPELQDWRULFV0DWKHPDWLFDO6SHFWUXP&micro;\$SSOLHG3URED-
ELOLW\7UXVWGLVSRQtYHOHPhttp://ms.appliedprobability.org/data/
&gt;@ :KLWZRUWK : \$ Choice and Chance (5th edition),
&gt;@0RUJDGR\$&amp;2&amp;DUYDOKR-%3&amp;DUYDOKR3&amp;3
&amp;RQVXOWiPRV DLQGD RV VLWHV %LE0#WKQHW www.bibmath.net)
&Agrave;OHVVHOHFWHGDUWLFOHVSGI
'HLJKWRQ%HOODQG&amp;R&amp;DPEULGJH
e MacTutor History of Mathematics (www-history.mcs.st-and.
ac.ukHVWH~OWLPRSDUDDVELRJUD&Agrave;DVGRVYiULRVPDWHPiWLFRV
referidos neste trabalho.
&gt;@0XLU7&acute;1RWHRQ6HOHFWHG&amp;RPELQDWLRQV&micro;Proceedings of the Royal Society of Edinburgh9RO;;,91HLO
DQG&amp;RPSDQ\/LPLWHG(GLQEXUJK
&gt;@1HYHV0\$)H*XHUUHLUR/Acesso ao Ensino Su-
SOBRE O AUTOR
perior – Matem&aacute;tica A3RUWR(GLWRUD3RUWR
Ant&oacute;nio Pereira Rosa &eacute; licenciado em Matem&aacute;tica (1986) e mestre em
Matem&aacute;tica para o Ensino (2008) pela Faculdade de Ci&ecirc;ncias da Universidade de Lisboa. &Eacute; professor do quadro da Escola Secund&aacute;ria Maria Am&aacute;lia Vaz
de Carvalho desde 1994.
&gt;@5\VHU+%Combinatorial Mathematics (The Carus
0DWKHPDWLFDO0RQRJUDSKV1RQGLPSUHVVLRQ, The Mathe-
PDWLFDO\$VVRFLDWLRQRI\$PHULFDGLVWULEXtGRSRU-RKQ:LOH\
DQG6RQV
&gt;@7RGKXQWHU,A History of the Mathematical Theory of
Probability0DF0LOODQDQG&amp;R&amp;DPEULGJH
&gt;@:DWVRQ5&acute;/RWWHUDO7KLQNLQJ&micro;Teaching StatisticsYROQ1RWWLQJKDP*Um%UHWDQKD
7
Os n&uacute;meros de Lucas s&atilde;o os termos da sucess&atilde;o 1, 3, 4, 7, 11, 18, 29, …,
definida por L1 = 1, L2 = 3, Ln = Ln−1 + Ln−2 (n &gt; 2). Estes n&uacute;meros gozam
de muitas propriedades semelhantes &agrave;s dos n&uacute;meros de Fibonacci 1, 1, 2, 3,
5, 8, 13, 21, …., j&aacute; que a rela&ccedil;&atilde;o de recorr&ecirc;ncia que os define &eacute; igual; apenas a
segunda condi&ccedil;&atilde;o inicial &eacute; diferente.
COELHOS E LOTARIAS rAnt&oacute;nio Pereira Rosa
23
```