+0  
 
0
526
7
avatar+14537 

Wie viele Möglichkeiten gibt es, den Geldbetrag von einem Euro ( 1 € ) nur mit Münzen von  1 , 2 oder  5 Cent zu bezahlen,  und wie kann man das berechnen ?

 16.08.2014

Beste Antwort 

 #7
avatar+3092 
+3

$$\left(\sum\limits_{i=1}^{100} x^{(1*i)} \right) \times \left(\sum\limits_{i=1}^{50} x^{(2*i)} \right) \times \left(\sum\limits_{i=1}^{20} x^{(5*i)} \right)$$

ergibt ausmultipliziert:

 

x^300+ x^299+ 2*x^298+ 2*x^297+ 3*x^296+ 4*x^295+ 5*x^294+ 6*x^293+ 7*x^292+ 8*x^291+ 10*x^290+ 11*x^289+ 13*x^288+ 14*x^287+ 16*x^286+ 18*x^285+ 20*x^284+ 22*x^283+ 24*x^282+ 26*x^281+ 29*x^280+ 31*x^279+ 34*x^278+ 36*x^277+ 39*x^276+ 42*x^275+ 45*x^274+ 48*x^273+ 51*x^272+ 54*x^271+ 58*x^270+ 61*x^269+ 65*x^268+ 68*x^267+ 72*x^266+ 76*x^265+ 80*x^264+ 84*x^263+ 88*x^262+ 92*x^261+ 97*x^260+ 101*x^259+ 106*x^258+ 110*x^257+ 115*x^256+ 120*x^255+ 125*x^254+ 130*x^253+ 135*x^252+ 140*x^251+ 146*x^250+ 151*x^249+ 157*x^248+ 162*x^247+ 168*x^246+ 174*x^245+ 180*x^244+ 186*x^243+ 192*x^242+ 198*x^241+ 205*x^240+ 211*x^239+ 218*x^238+ 224*x^237+ 231*x^236+ 238*x^235+ 245*x^234+ 252*x^233+ 259*x^232+ 266*x^231+ 274*x^230+ 281*x^229+ 289*x^228+ 296*x^227+ 304*x^226+ 312*x^225+ 320*x^224+ 328*x^223+ 336*x^222+ 344*x^221+ 353*x^220+ 361*x^219+ 370*x^218+ 378*x^217+ 387*x^216+ 396*x^215+ 405*x^214+ 414*x^213+ 423*x^212+ 432*x^211+ 442*x^210+ 451*x^209+ 461*x^208+ 470*x^207+ 480*x^206+ 490*x^205+ 500*x^204+ 510*x^203+ 520*x^202+ 530*x^201+ 541*x^200+ 550*x^199+ 560*x^198+ 569*x^197+ 579*x^196+ 588*x^195+ 597*x^194+ 605*x^193+ 614*x^192+ 622*x^191+ 631*x^190+ 638*x^189+ 646*x^188+ 653*x^187+ 661*x^186+ 668*x^185+ 675*x^184+ 681*x^183+ 688*x^182+ 694*x^181+ 701*x^180+ 706*x^179+ 712*x^178+ 717*x^177+ 723*x^176+ 728*x^175+ 733*x^174+ 737*x^173+ 742*x^172+ 746*x^171+ 751*x^170+ 754*x^169+ 758*x^168+ 761*x^167+ 765*x^166+ 768*x^165+ 771*x^164+ 773*x^163+ 776*x^162+ 778*x^161+ 781*x^160+ 782*x^159+ 784*x^158+ 785*x^157+ 787*x^156+ 788*x^155+ 789*x^154+ 789*x^153+ 790*x^152+ 790*x^151+ 791*x^150+ 790*x^149+ 790*x^148+ 789*x^147+ 789*x^146+ 788*x^145+ 787*x^144+ 785*x^143+ 784*x^142+ 782*x^141+ 781*x^140+ 778*x^139+ 776*x^138+ 773*x^137+ 771*x^136+ 768*x^135+ 765*x^134+ 761*x^133+ 758*x^132+ 754*x^131+ 751*x^130+ 746*x^129+ 742*x^128+ 737*x^127+ 733*x^126+ 728*x^125+ 723*x^124+ 717*x^123+ 712*x^122+ 706*x^121+ 701*x^120+ 694*x^119+ 688*x^118+ 681*x^117+ 675*x^116+ 668*x^115+ 661*x^114+ 653*x^113+ 646*x^112+ 638*x^111+ 631*x^110+ 622*x^109+ 614*x^108+ 605*x^107+ 597*x^106+ 588*x^105+ 579*x^104+ 569*x^103+ 560*x^102+ 550*x^101+ 541*x^100+ 530*x^99+ 520*x^98+ 510*x^97+ 500*x^96+ 490*x^95+ 480*x^94+ 470*x^93+ 461*x^92+ 451*x^91+ 442*x^90+ 432*x^89+ 423*x^88+ 414*x^87+ 405*x^86+ 396*x^85+ 387*x^84+ 378*x^83+ 370*x^82+ 361*x^81+ 353*x^80+ 344*x^79+ 336*x^78+ 328*x^77+ 320*x^76+ 312*x^75+ 304*x^74+ 296*x^73+ 289*x^72+ 281*x^71+ 274*x^70+ 266*x^69+ 259*x^68+ 252*x^67+ 245*x^66+ 238*x^65+ 231*x^64+ 224*x^63+ 218*x^62+ 211*x^61+ 205*x^60+ 198*x^59+ 192*x^58+ 186*x^57+ 180*x^56+ 174*x^55+ 168*x^54+ 162*x^53+ 157*x^52+ 151*x^51+ 146*x^50+ 140*x^49+ 135*x^48+ 130*x^47+ 125*x^46+ 120*x^45+ 115*x^44+ 110*x^43+ 106*x^42+ 101*x^41+ 97*x^40+ 92*x^39+ 88*x^38+ 84*x^37+ 80*x^36+ 76*x^35+ 72*x^34+ 68*x^33+ 65*x^32+ 61*x^31+ 58*x^30+ 54*x^29+ 51*x^28+ 48*x^27+ 45*x^26+ 42*x^25+ 39*x^24+ 36*x^23+ 34*x^22+ 31*x^21+ 29*x^20+ 26*x^19+ 24*x^18+ 22*x^17+ 20*x^16+ 18*x^15+ 16*x^14+ 14*x^13+ 13*x^12+ 11*x^11+ 10*x^10+ 8*x^9+ 7*x^8+ 6*x^7+ 5*x^6+ 4*x^5+ 3*x^4+ 2*x^3+ 2*x^2+ x+ 1

 

Der Koeffizient von x^100 ist 541. Also gibt es 541 Möglichkeiten.

--------------

siehe http://de.wikipedia.org/wiki/Erzeugende_Funktion

Oder: Mit Geometrischer Reihenformel die "Erzeugende Funktion" aufstellen:

(1/(1-x)) * (1/(1-x^2)) * (1/(1-x^5))

danach einfach die Taylorreihenentwicklung der 100. Ordnung berechnen:

 

1+ x+ 2*x^2+ 2*x^3+ 3*x^4+ 4*x^5+ 5*x^6+ 6*x^7+ 7*x^8+ 8*x^9+ 10*x^10+ 11*x^11+ 13*x^12+ 14*x^13+ 16*x^14+ 18*x^15+ 20*x^16+ 22*x^17+ 24*x^18+ 26*x^19+ 29*x^20+ 31*x^21+ 34*x^22+ 36*x^23+ 39*x^24+ 42*x^25+ 45*x^26+ 48*x^27+ 51*x^28+ 54*x^29+ 58*x^30+ 61*x^31+ 65*x^32+ 68*x^33+ 72*x^34+ 76*x^35+ 80*x^36+ 84*x^37+ 88*x^38+ 92*x^39+ 97*x^40+ 101*x^41+ 106*x^42+ 110*x^43+ 115*x^44+ 120*x^45+ 125*x^46+ 130*x^47+ 135*x^48+ 140*x^49+ 146*x^50+ 151*x^51+ 157*x^52+ 162*x^53+ 168*x^54+ 174*x^55+ 180*x^56+ 186*x^57+ 192*x^58+ 198*x^59+ 205*x^60+ 211*x^61+ 218*x^62+ 224*x^63+ 231*x^64+ 238*x^65+ 245*x^66+ 252*x^67+ 259*x^68+ 266*x^69+ 274*x^70+ 281*x^71+ 289*x^72+ 296*x^73+ 304*x^74+ 312*x^75+ 320*x^76+ 328*x^77+ 336*x^78+ 344*x^79+ 353*x^80+ 361*x^81+ 370*x^82+ 378*x^83+ 387*x^84+ 396*x^85+ 405*x^86+ 414*x^87+ 423*x^88+ 432*x^89+ 442*x^90+ 451*x^91+ 461*x^92+ 470*x^93+ 480*x^94+ 490*x^95+ 500*x^96+ 510*x^97+ 520*x^98+ 530*x^99+ 541*x^100+ O(x^101)

 17.08.2014
 #1
avatar+8142 
0

 

Hallo radix,

Ich wollte versuchen, über die Auflistung aller Zahlungsmöglichkeiten, zu einer Gesetzmäßigkeit zu kommen. Bisher tappe ich da noch im Dunkeln. Hier die Auflistung:

2*50+0*20+0*1

1*50+2*20+10*1

1*50+1*20+30*1

0*50+5*20+0*1

0*50+4*20+20*1

0*50+3*20+40*1

0*50+2*20+60*1

0*50+1*20+80*1

0*50+0*20+100*1

Das sind also 9 Möglichkeiten. Ich freue mich auf die mathematische Auflösung, die sicher von einem anderen Teilnehmer am Forum herausgefunden wird. Vielleicht komme ich auch selber noch drauf.

Gruß asinus  :- )

 16.08.2014
 #2
avatar+14537 
0

Hallo "Asinus",


vielen Dank für deinen ersten Versuch. Ich selber weiß auch nicht, wie man das berechnen kann.


Durch Auflistung wird man wohl nicht zum Ziel kommen, denn die Möglichkeiten sollen sich im Bereich   von  500  bewegen!


Gruß radix !

 16.08.2014
 #3
avatar+8142 
0

Hallo radix, ich habe mich verlesen und die Liste für 50ct, 20ct und 1ct aufgeschrieben.

Sorry! Gruß asinus  :- (

 16.08.2014
 #4
avatar+3092 
0

gesamt:541 Möglichkeiten
-----------------------------------------------
0x (1cent) + 0x (2cents) + 20x (5cents)
0x (1cent) + 5x (2cents) + 18x (5cents)
0x (1cent) + 10x (2cents) + 16x (5cents)
0x (1cent) + 15x (2cents) + 14x (5cents)
0x (1cent) + 20x (2cents) + 12x (5cents)
0x (1cent) + 25x (2cents) + 10x (5cents)
0x (1cent) + 30x (2cents) + 8x (5cents)
0x (1cent) + 35x (2cents) + 6x (5cents)
0x (1cent) + 40x (2cents) + 4x (5cents)
0x (1cent) + 45x (2cents) + 2x (5cents)
0x (1cent) + 50x (2cents) + 0x (5cents)
1x (1cent) + 2x (2cents) + 19x (5cents)
1x (1cent) + 7x (2cents) + 17x (5cents)
1x (1cent) + 12x (2cents) + 15x (5cents)
1x (1cent) + 17x (2cents) + 13x (5cents)
1x (1cent) + 22x (2cents) + 11x (5cents)
1x (1cent) + 27x (2cents) + 9x (5cents)
1x (1cent) + 32x (2cents) + 7x (5cents)
1x (1cent) + 37x (2cents) + 5x (5cents)
1x (1cent) + 42x (2cents) + 3x (5cents)
1x (1cent) + 47x (2cents) + 1x (5cents)
2x (1cent) + 4x (2cents) + 18x (5cents)
2x (1cent) + 9x (2cents) + 16x (5cents)
2x (1cent) + 14x (2cents) + 14x (5cents)
2x (1cent) + 19x (2cents) + 12x (5cents)
2x (1cent) + 24x (2cents) + 10x (5cents)
2x (1cent) + 29x (2cents) + 8x (5cents)
2x (1cent) + 34x (2cents) + 6x (5cents)
2x (1cent) + 39x (2cents) + 4x (5cents)
2x (1cent) + 44x (2cents) + 2x (5cents)
2x (1cent) + 49x (2cents) + 0x (5cents)
3x (1cent) + 1x (2cents) + 19x (5cents)
3x (1cent) + 6x (2cents) + 17x (5cents)
3x (1cent) + 11x (2cents) + 15x (5cents)
3x (1cent) + 16x (2cents) + 13x (5cents)
3x (1cent) + 21x (2cents) + 11x (5cents)
3x (1cent) + 26x (2cents) + 9x (5cents)
3x (1cent) + 31x (2cents) + 7x (5cents)
3x (1cent) + 36x (2cents) + 5x (5cents)
3x (1cent) + 41x (2cents) + 3x (5cents)
3x (1cent) + 46x (2cents) + 1x (5cents)
4x (1cent) + 3x (2cents) + 18x (5cents)
4x (1cent) + 8x (2cents) + 16x (5cents)
4x (1cent) + 13x (2cents) + 14x (5cents)
4x (1cent) + 18x (2cents) + 12x (5cents)
4x (1cent) + 23x (2cents) + 10x (5cents)
4x (1cent) + 28x (2cents) + 8x (5cents)
4x (1cent) + 33x (2cents) + 6x (5cents)
4x (1cent) + 38x (2cents) + 4x (5cents)
4x (1cent) + 43x (2cents) + 2x (5cents)
4x (1cent) + 48x (2cents) + 0x (5cents)
5x (1cent) + 0x (2cents) + 19x (5cents)
5x (1cent) + 5x (2cents) + 17x (5cents)
5x (1cent) + 10x (2cents) + 15x (5cents)
5x (1cent) + 15x (2cents) + 13x (5cents)
5x (1cent) + 20x (2cents) + 11x (5cents)
5x (1cent) + 25x (2cents) + 9x (5cents)
5x (1cent) + 30x (2cents) + 7x (5cents)
5x (1cent) + 35x (2cents) + 5x (5cents)
5x (1cent) + 40x (2cents) + 3x (5cents)
5x (1cent) + 45x (2cents) + 1x (5cents)
6x (1cent) + 2x (2cents) + 18x (5cents)
6x (1cent) + 7x (2cents) + 16x (5cents)
6x (1cent) + 12x (2cents) + 14x (5cents)
6x (1cent) + 17x (2cents) + 12x (5cents)
6x (1cent) + 22x (2cents) + 10x (5cents)
6x (1cent) + 27x (2cents) + 8x (5cents)
6x (1cent) + 32x (2cents) + 6x (5cents)
6x (1cent) + 37x (2cents) + 4x (5cents)
6x (1cent) + 42x (2cents) + 2x (5cents)
6x (1cent) + 47x (2cents) + 0x (5cents)
7x (1cent) + 4x (2cents) + 17x (5cents)
7x (1cent) + 9x (2cents) + 15x (5cents)
7x (1cent) + 14x (2cents) + 13x (5cents)
7x (1cent) + 19x (2cents) + 11x (5cents)
7x (1cent) + 24x (2cents) + 9x (5cents)
7x (1cent) + 29x (2cents) + 7x (5cents)
7x (1cent) + 34x (2cents) + 5x (5cents)
7x (1cent) + 39x (2cents) + 3x (5cents)
7x (1cent) + 44x (2cents) + 1x (5cents)
8x (1cent) + 1x (2cents) + 18x (5cents)
8x (1cent) + 6x (2cents) + 16x (5cents)
8x (1cent) + 11x (2cents) + 14x (5cents)
8x (1cent) + 16x (2cents) + 12x (5cents)
8x (1cent) + 21x (2cents) + 10x (5cents)
8x (1cent) + 26x (2cents) + 8x (5cents)
8x (1cent) + 31x (2cents) + 6x (5cents)
8x (1cent) + 36x (2cents) + 4x (5cents)
8x (1cent) + 41x (2cents) + 2x (5cents)
8x (1cent) + 46x (2cents) + 0x (5cents)
9x (1cent) + 3x (2cents) + 17x (5cents)
9x (1cent) + 8x (2cents) + 15x (5cents)
9x (1cent) + 13x (2cents) + 13x (5cents)
9x (1cent) + 18x (2cents) + 11x (5cents)
9x (1cent) + 23x (2cents) + 9x (5cents)
9x (1cent) + 28x (2cents) + 7x (5cents)
9x (1cent) + 33x (2cents) + 5x (5cents)
9x (1cent) + 38x (2cents) + 3x (5cents)
9x (1cent) + 43x (2cents) + 1x (5cents)
10x (1cent) + 0x (2cents) + 18x (5cents)
10x (1cent) + 5x (2cents) + 16x (5cents)
10x (1cent) + 10x (2cents) + 14x (5cents)
10x (1cent) + 15x (2cents) + 12x (5cents)
10x (1cent) + 20x (2cents) + 10x (5cents)
10x (1cent) + 25x (2cents) + 8x (5cents)
10x (1cent) + 30x (2cents) + 6x (5cents)
10x (1cent) + 35x (2cents) + 4x (5cents)
10x (1cent) + 40x (2cents) + 2x (5cents)
10x (1cent) + 45x (2cents) + 0x (5cents)
11x (1cent) + 2x (2cents) + 17x (5cents)
11x (1cent) + 7x (2cents) + 15x (5cents)
11x (1cent) + 12x (2cents) + 13x (5cents)
11x (1cent) + 17x (2cents) + 11x (5cents)
11x (1cent) + 22x (2cents) + 9x (5cents)
11x (1cent) + 27x (2cents) + 7x (5cents)
11x (1cent) + 32x (2cents) + 5x (5cents)
11x (1cent) + 37x (2cents) + 3x (5cents)
11x (1cent) + 42x (2cents) + 1x (5cents)
12x (1cent) + 4x (2cents) + 16x (5cents)
12x (1cent) + 9x (2cents) + 14x (5cents)
12x (1cent) + 14x (2cents) + 12x (5cents)
12x (1cent) + 19x (2cents) + 10x (5cents)
12x (1cent) + 24x (2cents) + 8x (5cents)
12x (1cent) + 29x (2cents) + 6x (5cents)
12x (1cent) + 34x (2cents) + 4x (5cents)
12x (1cent) + 39x (2cents) + 2x (5cents)
12x (1cent) + 44x (2cents) + 0x (5cents)
13x (1cent) + 1x (2cents) + 17x (5cents)
13x (1cent) + 6x (2cents) + 15x (5cents)
13x (1cent) + 11x (2cents) + 13x (5cents)
13x (1cent) + 16x (2cents) + 11x (5cents)
13x (1cent) + 21x (2cents) + 9x (5cents)
13x (1cent) + 26x (2cents) + 7x (5cents)
13x (1cent) + 31x (2cents) + 5x (5cents)
13x (1cent) + 36x (2cents) + 3x (5cents)
13x (1cent) + 41x (2cents) + 1x (5cents)
14x (1cent) + 3x (2cents) + 16x (5cents)
14x (1cent) + 8x (2cents) + 14x (5cents)
14x (1cent) + 13x (2cents) + 12x (5cents)
14x (1cent) + 18x (2cents) + 10x (5cents)
14x (1cent) + 23x (2cents) + 8x (5cents)
14x (1cent) + 28x (2cents) + 6x (5cents)
14x (1cent) + 33x (2cents) + 4x (5cents)
14x (1cent) + 38x (2cents) + 2x (5cents)
14x (1cent) + 43x (2cents) + 0x (5cents)
15x (1cent) + 0x (2cents) + 17x (5cents)
15x (1cent) + 5x (2cents) + 15x (5cents)
15x (1cent) + 10x (2cents) + 13x (5cents)
15x (1cent) + 15x (2cents) + 11x (5cents)
15x (1cent) + 20x (2cents) + 9x (5cents)
15x (1cent) + 25x (2cents) + 7x (5cents)
15x (1cent) + 30x (2cents) + 5x (5cents)
15x (1cent) + 35x (2cents) + 3x (5cents)
15x (1cent) + 40x (2cents) + 1x (5cents)
16x (1cent) + 2x (2cents) + 16x (5cents)
16x (1cent) + 7x (2cents) + 14x (5cents)
16x (1cent) + 12x (2cents) + 12x (5cents)
16x (1cent) + 17x (2cents) + 10x (5cents)
16x (1cent) + 22x (2cents) + 8x (5cents)
16x (1cent) + 27x (2cents) + 6x (5cents)
16x (1cent) + 32x (2cents) + 4x (5cents)
16x (1cent) + 37x (2cents) + 2x (5cents)
16x (1cent) + 42x (2cents) + 0x (5cents)
17x (1cent) + 4x (2cents) + 15x (5cents)
17x (1cent) + 9x (2cents) + 13x (5cents)
17x (1cent) + 14x (2cents) + 11x (5cents)
17x (1cent) + 19x (2cents) + 9x (5cents)
17x (1cent) + 24x (2cents) + 7x (5cents)
17x (1cent) + 29x (2cents) + 5x (5cents)
17x (1cent) + 34x (2cents) + 3x (5cents)
17x (1cent) + 39x (2cents) + 1x (5cents)
18x (1cent) + 1x (2cents) + 16x (5cents)
18x (1cent) + 6x (2cents) + 14x (5cents)
18x (1cent) + 11x (2cents) + 12x (5cents)
18x (1cent) + 16x (2cents) + 10x (5cents)
18x (1cent) + 21x (2cents) + 8x (5cents)
18x (1cent) + 26x (2cents) + 6x (5cents)
18x (1cent) + 31x (2cents) + 4x (5cents)
18x (1cent) + 36x (2cents) + 2x (5cents)
18x (1cent) + 41x (2cents) + 0x (5cents)
19x (1cent) + 3x (2cents) + 15x (5cents)
19x (1cent) + 8x (2cents) + 13x (5cents)
19x (1cent) + 13x (2cents) + 11x (5cents)
19x (1cent) + 18x (2cents) + 9x (5cents)
19x (1cent) + 23x (2cents) + 7x (5cents)
19x (1cent) + 28x (2cents) + 5x (5cents)
19x (1cent) + 33x (2cents) + 3x (5cents)
19x (1cent) + 38x (2cents) + 1x (5cents)
20x (1cent) + 0x (2cents) + 16x (5cents)
20x (1cent) + 5x (2cents) + 14x (5cents)
20x (1cent) + 10x (2cents) + 12x (5cents)
20x (1cent) + 15x (2cents) + 10x (5cents)
20x (1cent) + 20x (2cents) + 8x (5cents)
20x (1cent) + 25x (2cents) + 6x (5cents)
20x (1cent) + 30x (2cents) + 4x (5cents)
20x (1cent) + 35x (2cents) + 2x (5cents)
20x (1cent) + 40x (2cents) + 0x (5cents)
21x (1cent) + 2x (2cents) + 15x (5cents)
21x (1cent) + 7x (2cents) + 13x (5cents)
21x (1cent) + 12x (2cents) + 11x (5cents)
21x (1cent) + 17x (2cents) + 9x (5cents)
21x (1cent) + 22x (2cents) + 7x (5cents)
21x (1cent) + 27x (2cents) + 5x (5cents)
21x (1cent) + 32x (2cents) + 3x (5cents)
21x (1cent) + 37x (2cents) + 1x (5cents)
22x (1cent) + 4x (2cents) + 14x (5cents)
22x (1cent) + 9x (2cents) + 12x (5cents)
22x (1cent) + 14x (2cents) + 10x (5cents)
22x (1cent) + 19x (2cents) + 8x (5cents)
22x (1cent) + 24x (2cents) + 6x (5cents)
22x (1cent) + 29x (2cents) + 4x (5cents)
22x (1cent) + 34x (2cents) + 2x (5cents)
22x (1cent) + 39x (2cents) + 0x (5cents)
23x (1cent) + 1x (2cents) + 15x (5cents)
23x (1cent) + 6x (2cents) + 13x (5cents)
23x (1cent) + 11x (2cents) + 11x (5cents)
23x (1cent) + 16x (2cents) + 9x (5cents)
23x (1cent) + 21x (2cents) + 7x (5cents)
23x (1cent) + 26x (2cents) + 5x (5cents)
23x (1cent) + 31x (2cents) + 3x (5cents)
23x (1cent) + 36x (2cents) + 1x (5cents)
24x (1cent) + 3x (2cents) + 14x (5cents)
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66x (1cent) + 12x (2cents) + 2x (5cents)
66x (1cent) + 17x (2cents) + 0x (5cents)
67x (1cent) + 4x (2cents) + 5x (5cents)
67x (1cent) + 9x (2cents) + 3x (5cents)
67x (1cent) + 14x (2cents) + 1x (5cents)
68x (1cent) + 1x (2cents) + 6x (5cents)
68x (1cent) + 6x (2cents) + 4x (5cents)
68x (1cent) + 11x (2cents) + 2x (5cents)
68x (1cent) + 16x (2cents) + 0x (5cents)
69x (1cent) + 3x (2cents) + 5x (5cents)
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76x (1cent) + 2x (2cents) + 4x (5cents)
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78x (1cent) + 1x (2cents) + 4x (5cents)
78x (1cent) + 6x (2cents) + 2x (5cents)
78x (1cent) + 11x (2cents) + 0x (5cents)
79x (1cent) + 3x (2cents) + 3x (5cents)
79x (1cent) + 8x (2cents) + 1x (5cents)
80x (1cent) + 0x (2cents) + 4x (5cents)
80x (1cent) + 5x (2cents) + 2x (5cents)
80x (1cent) + 10x (2cents) + 0x (5cents)
81x (1cent) + 2x (2cents) + 3x (5cents)
81x (1cent) + 7x (2cents) + 1x (5cents)
82x (1cent) + 4x (2cents) + 2x (5cents)
82x (1cent) + 9x (2cents) + 0x (5cents)
83x (1cent) + 1x (2cents) + 3x (5cents)
83x (1cent) + 6x (2cents) + 1x (5cents)
84x (1cent) + 3x (2cents) + 2x (5cents)
84x (1cent) + 8x (2cents) + 0x (5cents)
85x (1cent) + 0x (2cents) + 3x (5cents)
85x (1cent) + 5x (2cents) + 1x (5cents)
86x (1cent) + 2x (2cents) + 2x (5cents)
86x (1cent) + 7x (2cents) + 0x (5cents)
87x (1cent) + 4x (2cents) + 1x (5cents)
88x (1cent) + 1x (2cents) + 2x (5cents)
88x (1cent) + 6x (2cents) + 0x (5cents)
89x (1cent) + 3x (2cents) + 1x (5cents)
90x (1cent) + 0x (2cents) + 2x (5cents)
90x (1cent) + 5x (2cents) + 0x (5cents)
91x (1cent) + 2x (2cents) + 1x (5cents)
92x (1cent) + 4x (2cents) + 0x (5cents)
93x (1cent) + 1x (2cents) + 1x (5cents)
94x (1cent) + 3x (2cents) + 0x (5cents)
95x (1cent) + 0x (2cents) + 1x (5cents)
96x (1cent) + 2x (2cents) + 0x (5cents)
98x (1cent) + 1x (2cents) + 0x (5cents)
100x (1cent) + 0x (2cents) + 0x (5cents)

 16.08.2014
 #5
avatar+8142 
0

Hallo admin,

deine Liste ist eine grandiose Fleißleistung. Super!

Gruß asinus :- )

 16.08.2014
 #6
avatar+14537 
0

Hallo Asinus, hallo Admin,

recht herzlichen Dank für eure Bemühungen !

Da es eine Aufgabe für die 9. Klassenstufe war, müsste es wohl auch eine relativ einfache Berechnungsformel aus der Kombinatorik geben. Ich vermute, dass sie Admin gefunden hat und damit die lange Liste mit dem richtigen Ergebnis ( 541 ) ausgedruckt hat.

Noch einmal herzlichen Dank sagt radix !, der sich in dem Gebiet Kombinatorik leider nicht auskennt.

 

 16.08.2014
 #7
avatar+3092 
+3
Beste Antwort

$$\left(\sum\limits_{i=1}^{100} x^{(1*i)} \right) \times \left(\sum\limits_{i=1}^{50} x^{(2*i)} \right) \times \left(\sum\limits_{i=1}^{20} x^{(5*i)} \right)$$

ergibt ausmultipliziert:

 

x^300+ x^299+ 2*x^298+ 2*x^297+ 3*x^296+ 4*x^295+ 5*x^294+ 6*x^293+ 7*x^292+ 8*x^291+ 10*x^290+ 11*x^289+ 13*x^288+ 14*x^287+ 16*x^286+ 18*x^285+ 20*x^284+ 22*x^283+ 24*x^282+ 26*x^281+ 29*x^280+ 31*x^279+ 34*x^278+ 36*x^277+ 39*x^276+ 42*x^275+ 45*x^274+ 48*x^273+ 51*x^272+ 54*x^271+ 58*x^270+ 61*x^269+ 65*x^268+ 68*x^267+ 72*x^266+ 76*x^265+ 80*x^264+ 84*x^263+ 88*x^262+ 92*x^261+ 97*x^260+ 101*x^259+ 106*x^258+ 110*x^257+ 115*x^256+ 120*x^255+ 125*x^254+ 130*x^253+ 135*x^252+ 140*x^251+ 146*x^250+ 151*x^249+ 157*x^248+ 162*x^247+ 168*x^246+ 174*x^245+ 180*x^244+ 186*x^243+ 192*x^242+ 198*x^241+ 205*x^240+ 211*x^239+ 218*x^238+ 224*x^237+ 231*x^236+ 238*x^235+ 245*x^234+ 252*x^233+ 259*x^232+ 266*x^231+ 274*x^230+ 281*x^229+ 289*x^228+ 296*x^227+ 304*x^226+ 312*x^225+ 320*x^224+ 328*x^223+ 336*x^222+ 344*x^221+ 353*x^220+ 361*x^219+ 370*x^218+ 378*x^217+ 387*x^216+ 396*x^215+ 405*x^214+ 414*x^213+ 423*x^212+ 432*x^211+ 442*x^210+ 451*x^209+ 461*x^208+ 470*x^207+ 480*x^206+ 490*x^205+ 500*x^204+ 510*x^203+ 520*x^202+ 530*x^201+ 541*x^200+ 550*x^199+ 560*x^198+ 569*x^197+ 579*x^196+ 588*x^195+ 597*x^194+ 605*x^193+ 614*x^192+ 622*x^191+ 631*x^190+ 638*x^189+ 646*x^188+ 653*x^187+ 661*x^186+ 668*x^185+ 675*x^184+ 681*x^183+ 688*x^182+ 694*x^181+ 701*x^180+ 706*x^179+ 712*x^178+ 717*x^177+ 723*x^176+ 728*x^175+ 733*x^174+ 737*x^173+ 742*x^172+ 746*x^171+ 751*x^170+ 754*x^169+ 758*x^168+ 761*x^167+ 765*x^166+ 768*x^165+ 771*x^164+ 773*x^163+ 776*x^162+ 778*x^161+ 781*x^160+ 782*x^159+ 784*x^158+ 785*x^157+ 787*x^156+ 788*x^155+ 789*x^154+ 789*x^153+ 790*x^152+ 790*x^151+ 791*x^150+ 790*x^149+ 790*x^148+ 789*x^147+ 789*x^146+ 788*x^145+ 787*x^144+ 785*x^143+ 784*x^142+ 782*x^141+ 781*x^140+ 778*x^139+ 776*x^138+ 773*x^137+ 771*x^136+ 768*x^135+ 765*x^134+ 761*x^133+ 758*x^132+ 754*x^131+ 751*x^130+ 746*x^129+ 742*x^128+ 737*x^127+ 733*x^126+ 728*x^125+ 723*x^124+ 717*x^123+ 712*x^122+ 706*x^121+ 701*x^120+ 694*x^119+ 688*x^118+ 681*x^117+ 675*x^116+ 668*x^115+ 661*x^114+ 653*x^113+ 646*x^112+ 638*x^111+ 631*x^110+ 622*x^109+ 614*x^108+ 605*x^107+ 597*x^106+ 588*x^105+ 579*x^104+ 569*x^103+ 560*x^102+ 550*x^101+ 541*x^100+ 530*x^99+ 520*x^98+ 510*x^97+ 500*x^96+ 490*x^95+ 480*x^94+ 470*x^93+ 461*x^92+ 451*x^91+ 442*x^90+ 432*x^89+ 423*x^88+ 414*x^87+ 405*x^86+ 396*x^85+ 387*x^84+ 378*x^83+ 370*x^82+ 361*x^81+ 353*x^80+ 344*x^79+ 336*x^78+ 328*x^77+ 320*x^76+ 312*x^75+ 304*x^74+ 296*x^73+ 289*x^72+ 281*x^71+ 274*x^70+ 266*x^69+ 259*x^68+ 252*x^67+ 245*x^66+ 238*x^65+ 231*x^64+ 224*x^63+ 218*x^62+ 211*x^61+ 205*x^60+ 198*x^59+ 192*x^58+ 186*x^57+ 180*x^56+ 174*x^55+ 168*x^54+ 162*x^53+ 157*x^52+ 151*x^51+ 146*x^50+ 140*x^49+ 135*x^48+ 130*x^47+ 125*x^46+ 120*x^45+ 115*x^44+ 110*x^43+ 106*x^42+ 101*x^41+ 97*x^40+ 92*x^39+ 88*x^38+ 84*x^37+ 80*x^36+ 76*x^35+ 72*x^34+ 68*x^33+ 65*x^32+ 61*x^31+ 58*x^30+ 54*x^29+ 51*x^28+ 48*x^27+ 45*x^26+ 42*x^25+ 39*x^24+ 36*x^23+ 34*x^22+ 31*x^21+ 29*x^20+ 26*x^19+ 24*x^18+ 22*x^17+ 20*x^16+ 18*x^15+ 16*x^14+ 14*x^13+ 13*x^12+ 11*x^11+ 10*x^10+ 8*x^9+ 7*x^8+ 6*x^7+ 5*x^6+ 4*x^5+ 3*x^4+ 2*x^3+ 2*x^2+ x+ 1

 

Der Koeffizient von x^100 ist 541. Also gibt es 541 Möglichkeiten.

--------------

siehe http://de.wikipedia.org/wiki/Erzeugende_Funktion

Oder: Mit Geometrischer Reihenformel die "Erzeugende Funktion" aufstellen:

(1/(1-x)) * (1/(1-x^2)) * (1/(1-x^5))

danach einfach die Taylorreihenentwicklung der 100. Ordnung berechnen:

 

1+ x+ 2*x^2+ 2*x^3+ 3*x^4+ 4*x^5+ 5*x^6+ 6*x^7+ 7*x^8+ 8*x^9+ 10*x^10+ 11*x^11+ 13*x^12+ 14*x^13+ 16*x^14+ 18*x^15+ 20*x^16+ 22*x^17+ 24*x^18+ 26*x^19+ 29*x^20+ 31*x^21+ 34*x^22+ 36*x^23+ 39*x^24+ 42*x^25+ 45*x^26+ 48*x^27+ 51*x^28+ 54*x^29+ 58*x^30+ 61*x^31+ 65*x^32+ 68*x^33+ 72*x^34+ 76*x^35+ 80*x^36+ 84*x^37+ 88*x^38+ 92*x^39+ 97*x^40+ 101*x^41+ 106*x^42+ 110*x^43+ 115*x^44+ 120*x^45+ 125*x^46+ 130*x^47+ 135*x^48+ 140*x^49+ 146*x^50+ 151*x^51+ 157*x^52+ 162*x^53+ 168*x^54+ 174*x^55+ 180*x^56+ 186*x^57+ 192*x^58+ 198*x^59+ 205*x^60+ 211*x^61+ 218*x^62+ 224*x^63+ 231*x^64+ 238*x^65+ 245*x^66+ 252*x^67+ 259*x^68+ 266*x^69+ 274*x^70+ 281*x^71+ 289*x^72+ 296*x^73+ 304*x^74+ 312*x^75+ 320*x^76+ 328*x^77+ 336*x^78+ 344*x^79+ 353*x^80+ 361*x^81+ 370*x^82+ 378*x^83+ 387*x^84+ 396*x^85+ 405*x^86+ 414*x^87+ 423*x^88+ 432*x^89+ 442*x^90+ 451*x^91+ 461*x^92+ 470*x^93+ 480*x^94+ 490*x^95+ 500*x^96+ 510*x^97+ 520*x^98+ 530*x^99+ 541*x^100+ O(x^101)

admin 17.08.2014

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