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由0-9組成一個六位數，其中有兩個數字是相同的，除此之外，每個數字均不相同，這
六個數字的總和為30，問這樣的六位數有幾個？
（例如說，678900、478911都是）

真....真是個有挑戰性的題目=.=....

慢慢算....(土法鍊鋼中orz....

(2) 由0-9組成一個六元素有序數列，這六個數字的總和為30問這樣的有序數列有幾個？

F=(1+x+xx+x^3+...+x^9)^6
=(1-x^10)^6/(1-x)^6
=(1-6*x^10+15*x^20-20*x^30+...)/(1-x)^6

x^30項係數
=C(30+5,5)-6*C(25,5)+15C(15,5)-20C(5,5)
=50877

(3) 六位數六個數字的總和為30，問這樣的六位數有幾個?

F=(x+xx+...+x^9) (1+x+xx+x^3+...+x^9)^5
=x(1-x^9)(1-x^10)^5/(1-x)^6
=x(1-5*x^10+10*x^20-10*x^30+...)(1-x^9)/(1-x)^6
=x(1-x^9-5*x^10+5x^19+10*x^20-10x^29-10*x^30+...)/(1-x)^6

x^30項係數
=C(34,5)-C(25,5)-5*C(24,5)+5C(15,5)+10C(14,5)-10C(5,5)
=47631

----
or
F=(1+x+xx+x^3+...+x^9)^6-(1+x+xx+...+x^9)^5
=(1-x^10)^6/(1-x)^6 -(1-x^10)^5/(1-x)^5
=(1-x^10)^6/(1-x)^6 -(1-5*x^10+10*x^20-10*x^30+...)/(1-x)^5

x^30項係數
=50877-[C(34,4)-5*C(24,4)+10C(14,4)-10C(4,4)]
=47631

雞腿飯 寫到:(2) 由0-9組成一個六元素有序數列，這六個數字的總和為30問這樣的有序數列有幾個？

F=(1+x+xx+x^3+...+x^9)^6
=(1-x^10)^6/(1-x)^6
=(1-6*x^10+15*x^20-20*x^30+...)/(1-x)^6

x^30項係數
=C(30+5,5)-6*C(25,5)+15C(15,5)-20C(5,5)
=50877

(3) 六位數六個數字的總和為30，問這樣的六位數有幾個?

F=(x+xx+...+x^9) (1+x+xx+x^3+...+x^9)^5
=x(1-x^9)(1-x^10)^5/(1-x)^6
=x(1-5*x^10+10*x^20-10*x^30+...)(1-x^9)/(1-x)^6
=x(1-x^9-5*x^10+5x^19+10*x^20-10x^29-10*x^30+...)/(1-x)^6

x^30項係數
=C(34,5)-C(25,5)-5*C(24,5)+5C(15,5)+10C(14,5)-10C(5,5)
=47631

----
or
F=(1+x+xx+x^3+...+x^9)^6-(1+x+xx+...+x^9)^5
=(1-x^10)^6/(1-x)^6 -(1-x^10)^5/(1-x)^5
=(1-x^10)^6/(1-x)^6 -(1-5*x^10+10*x^20-10*x^30+...)/(1-x)^5

x^30項係數
=50877-[C(34,4)-5*C(24,4)+10C(14,4)-10C(4,4)]
=47631

六位數六個數字的總和為30，問這樣的六位數有幾個?
H(6,30)-C(6,1)*H(6,20)+C(6,2)*H(6,10)-C(6,3)*H(6,0).........(1)
H(5,30)-C(5,1)*H(5,20)+C(5,2)*H(5,10)-C(5,3)*H(5,0)..........(2)
(1)-(2)就是答案

30=0+0+6+7+8+9
=1+1+4+7+8+9
=1+1+5+6+8+9
=2+2+3+6+8+9
=2+2+4+5+8+9
=2+2+4+6+7+9
=2+2+5+6+7+8
=3+3+1+6+8+9
=3+3+2+5+8+9
=3+3+2+6+7+9
=3+3+4+5+7+8
=4+4+2+3+8+9
=4+4+1+5+7+9
=4+4+1+6+7+8
=4+4+2+5+7+8
=4+4+3+5+6+8
=5+5+1+2+8+9
=5+5+1+3+7+9
=5+5+1+4+7+8
=5+5+2+3+7+8
=5+5+2+4+6+8
=5+5+3+4+6+7
=6+6+2+3+4+9
=6+6+1+4+5+8
=6+6+2+3+5+8
=6+6+2+4+5+7
=7+7+1+2+5+8
=7+7+1+2+4+9
=7+7+1+3+4+8
=8+8+1+2+5+6
=8=8+1+2+4+7
=8+8+1+3+4+6
=8+8+2+3+4+5
=9+9+1+2+3+6


共240+37*6!/2!=13560
不知有沒有漏算

qeypour 寫到:

=8+8+2+3+4+5
=9+9+1+2+3+6


共240+37*6!/2!=13560
不知有沒有漏算

至少漏掉991245

qeypour 寫到:30=0+0+6+7+8+9
=1+1+4+7+8+9
=1+1+5+6+8+9
=2+2+3+6+8+9
=2+2+4+5+8+9
=2+2+4+6+7+9
=2+2+5+6+7+8
=3+3+1+6+8+9
=3+3+2+5+8+9
=3+3+2+6+7+9
=3+3+4+5+7+8
=4+4+2+3+8+9
=4+4+1+5+7+9
=4+4+1+6+7+8
=4+4+2+5+7+8
=4+4+3+5+6+8
=5+5+1+2+8+9
=5+5+1+3+7+9
=5+5+1+4+7+8
=5+5+2+3+7+8
=5+5+2+4+6+8
=5+5+3+4+6+7
=6+6+2+3+4+9
=6+6+1+4+5+8
=6+6+2+3+5+8
=6+6+2+4+5+7
=7+7+1+2+5+8
=7+7+1+2+4+9
=7+7+1+3+4+8
=8+8+1+2+5+6
=8=8+1+2+4+7
=8+8+1+3+4+6
=8+8+2+3+4+5
=9+9+1+2+3+6


共240+37*6!/2!=13560
不知有沒有漏算

至少漏寫551469,991245

路人DF 寫到:

qeypour 寫到:30=0+0+6+7+8+9
=1+1+4+7+8+9
=1+1+5+6+8+9
=2+2+3+6+8+9
=2+2+4+5+8+9
=2+2+4+6+7+9
=2+2+5+6+7+8
=3+3+1+6+8+9
=3+3+2+5+8+9
=3+3+2+6+7+9
=3+3+4+5+7+8
=4+4+2+3+8+9
=4+4+1+5+7+9
=4+4+1+6+7+8
=4+4+2+5+7+8
=4+4+3+5+6+8
=5+5+1+2+8+9
=5+5+1+3+7+9
=5+5+1+4+7+8
=5+5+2+3+7+8
=5+5+2+4+6+8
=5+5+3+4+6+7
=6+6+2+3+4+9
=6+6+1+4+5+8
=6+6+2+3+5+8
=6+6+2+4+5+7
=7+7+1+2+5+8
=7+7+1+2+4+9
=7+7+1+3+4+8
=8+8+1+2+5+6
=8=8+1+2+4+7
=8+8+1+3+4+6
=8+8+2+3+4+5
=9+9+1+2+3+6


共240+37*6!/2!=13560
不知有沒有漏算

至少漏寫551469,991245

個人覺得這題應交由電腦運算比較恰當
不然位數跟數字和都很大時應非人腦可負荷
不知J+W兄有無妙法快解?