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 #2
avatar+26396 
+15

We have 10 standard 6-sided dice, all different colors.

In how many ways can we roll them to get a sum of 20?

 

1.

The number of ways of partitioning 20 into exactly 10 distinct parts with  the 6 numbers {1,2,3,4,5,6} we have:

30 partitions of 20 into ten distinct parts:

 

The 30 distinct partitions with the sum 20 are:

   1.) {2,2,2,2,2,2,2,2,2,2}
   2.) {1,2,2,2,2,2,2,2,2,3}
   3.) {1,1,2,2,2,2,2,2,3,3}
   4.) {1,1,2,2,2,2,2,2,2,4}
   5.) {1,1,1,2,2,2,2,3,3,3} 
   6.) {1,1,1,2,2,2,2,2,3,4}
   7.) {1,1,1,2,2,2,2,2,2,5}
   8.) {1,1,1,1,2,2,3,3,3,3}
   9.) {1,1,1,1,2,2,2,3,3,4}
  10.) {1,1,1,1,2,2,2,2,4,4}
  11.) {1,1,1,1,2,2,2,2,3,5}
  12.) {1,1,1,1,2,2,2,2,2,6}
  13.) {1,1,1,1,1,3,3,3,3,3}
  14.) {1,1,1,1,1,2,3,3,3,4}
  15.) {1,1,1,1,1,2,2,3,4,4}
  16.) {1,1,1,1,1,2,2,3,3,5}
  17.) {1,1,1,1,1,2,2,2,4,5}
  18.) {1,1,1,1,1,2,2,2,3,6}
  19.) {1,1,1,1,1,1,3,3,4,4}
  20.) {1,1,1,1,1,1,3,3,3,5}
  21.) {1,1,1,1,1,1,2,4,4,4}
  22.) {1,1,1,1,1,1,2,3,4,5}
  23.) {1,1,1,1,1,1,2,3,3,6}
  24.) {1,1,1,1,1,1,2,2,5,5}
  25.) {1,1,1,1,1,1,2,2,4,6}
  26.) {1,1,1,1,1,1,1,4,4,5}
  27.) {1,1,1,1,1,1,1,3,5,5}
  28.) {1,1,1,1,1,1,1,3,4,6}
  29.) {1,1,1,1,1,1,1,2,5,6}
  30.) {1,1,1,1,1,1,1,1,6,6}

 

2.

In how many ways can we roll them to get a sum of 20?
We need the permutations of each partitions and the sum is the answer.

1.){2,2,2,2,2,2,2,2,2,2}10!10!=12.){1,2,2,2,2,2,2,2,2,3}10!1!8!1!=903.){1,1,2,2,2,2,2,2,3,3}10!2!6!2!=12604.){1,1,2,2,2,2,2,2,2,4}10!2!6!1!=3605.){1,1,1,2,2,2,2,3,3,3}10!3!4!3!=42006.){1,1,1,2,2,2,2,2,3,4}10!3!5!1!1!=50407.){1,1,1,2,2,2,2,2,2,5}10!3!6!1!=8408.){1,1,1,1,2,2,3,3,3,3}10!4!2!4!=31509.){1,1,1,1,2,2,2,3,3,4}10!4!3!2!1!=1260010.){1,1,1,1,2,2,2,2,4,4}10!4!4!2!=315011.){1,1,1,1,2,2,2,2,3,5}10!4!4!1!1!=630012.){1,1,1,1,2,2,2,2,2,6}10!4!5!1!=126013.){1,1,1,1,1,3,3,3,3,3}10!5!5!=25214.){1,1,1,1,1,2,3,3,3,4}10!5!1!3!1!=504015.){1,1,1,1,1,2,2,3,4,4}10!5!2!1!2!=756016.){1,1,1,1,1,2,2,3,3,5}10!5!2!2!1!=756017.){1,1,1,1,1,2,2,2,4,5}10!5!3!1!1!=504018.){1,1,1,1,1,2,2,2,3,6}10!5!3!1!1!=504019.){1,1,1,1,1,1,3,3,4,4}10!6!2!2!=126020.){1,1,1,1,1,1,3,3,3,5}10!6!3!1!=84021.){1,1,1,1,1,1,2,4,4,4}10!6!1!3!=84022.){1,1,1,1,1,1,2,3,4,5}10!6!1!1!1!1!=504023.){1,1,1,1,1,1,2,3,3,6}10!6!1!2!1!=252024.){1,1,1,1,1,1,2,2,5,5}10!6!2!2!=126025.){1,1,1,1,1,1,2,2,4,6}10!6!2!1!1!=252026.){1,1,1,1,1,1,1,4,4,5}10!7!2!1!=36027.){1,1,1,1,1,1,1,3,5,5}10!7!1!2!=36028.){1,1,1,1,1,1,1,3,4,6}10!7!1!1!1!=72029.){1,1,1,1,1,1,1,2,5,6}10!7!1!1!1!=72030.){1,1,1,1,1,1,1,1,6,6}10!8!2!=45sum=85228

 

 

3.

The probability is:

 

85228610=8522860466176=0.00140951530

 

laugh

10.05.2016
 #2
avatar+26396 
+10

log25+log2xlog50+(log2)^2=?

 

log(25)+log(2)log(50)+[log(2)]2=log(52)+log(2)log(252)+log(2)log(2)=2log(5)+log(2)[ log(2)+log(52) ]+log(2)log(2)=2log(5)+log(2)log(2)+log(2)log(52)+log(2)log(2)=2log(5)+2log(2)log(2)+log(2)log(52)=2log(5)+2log(2)log(2)+2log(2)log(5)=2log(5)+2log(2)[ log(2)+log(5) ]=2log(5)+2log(2)log(25)=2log(5)+2log(2)log(10)|log10(10)=1=2log(5)+2log(2)1=2log(5)+2log(2)=2[ log(5)+log(2) ]=2log(52)=2log(10)|log10(10)=1=21=2log(25)+log(2)log(50)+[log(2)]2=2

laugh

.
10.05.2016
 #3
avatar+26396 
+5

Observe the pattern: 2,5,8,14,... if the pattern continues, what is the 41st number?

 

Primes of the form 4n-1 p3p1413a1=3314=227a2=3714=5311a3=31114=8419a4=31914=14523a5=32314=17631237433284735959441067501171531279591383621410377151078016127951713198181391041915111320163122211671252217913423191143241991492521115826223167272271702823917929251188302631973127120332283212333072303431123335331248363472603735926938367275393792844038328741419a41=341914=31442431323434393294444333245463347

 

laugh

09.05.2016
 #1
avatar+26396 
+12

Evaluate without a calculator:

[ 12(33)+11(43)+10(53)++2(133)+(143) ]

 

Identity:

For  n,rN+,rn,(n+1r+1)=nj=r(jr)

see: http://www.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Pascal's_triangle.html

 

(33)+(43)+(53)+(63)+(73)+(83)+(93)+(103)+(113)+(123)+(133)+(143)=(154)(33)+(43)+(53)+(63)+(73)+(83)+(93)+(103)+(113)+(123)+(133)=(144)(33)+(43)+(53)+(63)+(73)+(83)+(93)+(103)+(113)+(123)=(134)(33)+(43)+(53)+(63)+(73)+(83)+(93)+(103)+(113)=(124)(33)+(43)+(53)+(63)+(73)+(83)+(93)+(103)=(114)(33)+(43)+(53)+(63)+(73)+(83)+(93)=(104)(33)+(43)+(53)+(63)+(73)+(83)=(94)(33)+(43)+(53)+(63)+(73)=(84)(33)+(43)+(53)+(63)=(74)(33)+(43)+(53)=(64)(33)+(43)=(54)(33)=(44)

 

(44)+(54)+(64)+(74)+(84)+(94)+(104)+(114)+(124)+(134)+(144)+(154)=(165)12(33)+11(43)+10(53)+9(63)+8(73)+7(83)+6(93)+5(103)+4(113)+3(123)+2(133)+(143)=(165)12(33)+11(43)+10(53)+9(63)+8(73)+7(83)+6(93)+5(103)+4(113)+3(123)+2(133)+(143)=16515414313212112(33)+11(43)+10(53)+9(63)+8(73)+7(83)+6(93)+5(103)+4(113)+3(123)+2(133)+(143)=4368

.
09.05.2016
 #2
avatar+26396 
+5

if i have sides with each having a length of 13, 14, and 15. how big will each angle be?

 

 

Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is
A=s(sa)(sb)(sc),
where s is the semiperimeter of the triangle; that is,
s=a+b+c2

 

s=a+b+c2=13+14+152=422s=21A=s(sa)(sb)(sc)=21(2113)(2114)(2115)=21876=7056A=84

 

2A=bcsinαsinα=2Abcsinα=2841415α=arcsin(1681415)α=arcsin(0.8)α=53.1301023542

 

2A=casinβsinβ=2Acasinβ=2841513β=arcsin(1681513)β=arcsin(0.86153846154)β=59.4897625939

 

2A=absinγsinγ=2Aabsinγ=2841314γ=arcsin(1681314)γ=arcsin(0.92307692308)γ=67.3801350520

 

laugh

09.05.2016
 #1
avatar+26396 
0

Bitte um Hilfe: Wie kann man folgende Angabe mit einer Gleichung lösen?

In einem Geschäft gibt es blaue, grüne, weiße und schwarze Pullover. Es sind halb so viele blaue wie grüne Pullover. Es sind halb so viele grüne wie weiße Pullover. Von den schwarzen Pullovern gibt es 18 Stück, das sind halb so viele wie von den grünen und weißen Pullovern zusammen. Wie viele Pullover gibt es in diesem Geschäft?

Ich habe zwar die Lösung von der Anzahl der Pullovers. Ich weiß aber leider nicht, wie ich mit einer oder zwei Gleichungen diese Aufgabe lösen kann. Bitte um dringende Hilfe! Vielen Dank!!!

 

b = blau

g = grün

w = weiß

s = schwarz

 

(1)g=2b(2)w=2gw=22bw=4b(3)s=18(4)g+w=2sg+w=218g+w=36(5)g+w=2b+4bg+w=6b(4)=(5)g+w=36g+w=6b6b=36b=366b=6g=2bg=26g=12w=2gw=212w=24b+g+w+s=6+12+24+18=60b+g+w+s=60

 

Die Anzahl der Pullover sind 60.

 

Es gibt 6 blaue Pullover.

Es gibt 12 grüne Pullover.

Es gibt 24 weiße Pullover.

Es gibt 18 schwarze Pullover.

 

laugh

09.05.2016