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)
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
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
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