有一種四位數abcd, 其中a>b,b<c,c>d; 例如7350,1061,9582試問此種四位數有幾個?

當 b = 0

a = 1 ~ 9

當 c = 1 , d = 0 , 有 1 組解

當 c = 2 , d = 0 ~ 1 , 有 2 組解

.........

當 c = 9 , d = 0 ~ 8 , 有 9 組解

因為 c,d 的組合由 b 決定, 與 a 無關 , 所以:

可能組數

= a的可能數 * (c,d)的組合數

= 9 * ( 1 + 2 + ..... + 9 )

= 9 * 9(1+9)/2

= 9^2 * 10 / 2

當 b = 1

a = 2 ~ 9

當 c = 2 , d = 0 ~ 1 , 有 2 組解

當 c = 3 , d = 0 ~ 2 , 有 3 組解

.........

當 c = 9 , d = 0 ~ 8 , 有 9 組解

可能組數

= a的可能數 * (c,d)的組合數

= 8 * ( 2 + 3 + ..... + 9 )

= 8 * 8(2+9)/2

= 8^2 * 11 / 2

當 b = 2 ~ 7 , 與以上類似, 故省略

當 b = 8

a = 9 , c = 9 , d = 0 ~ 8

可能組數 = 9

所有可能組數

= (9^2)*10/2 + (8^2)*11/2 + ..... + (1^2)*18/2

= (1/2) * Σ (k^2)*( 19 - k ) , from k = 1 to k = 9

= (1/2) * [ 19*Σ k^2 - Σ k^3 ]

= (19/2)*Σ k^2 - (1/2)*Σ k^3

= (19/2)*n(n+1)(2n+1)/6 - (1/2)*[ n(n+1)/2 ]^2 , where n = 9

= (19/2)*9*10*19/6 - (1/2)( 9*10/2 )^2

= 2705.5 - 1012.5

= 1695

Ans: 1695 個

驗證:

可以用程式驗證, 以下為 Excel VBA 程式碼:

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Sub Ex()

S = 0

For n = 1000 To 9999

T = n

d = T Mod 10

T = Int(T / 10)

c = T Mod 10

T = Int(T / 10)

b = T Mod 10

T = Int(T / 10)

a = T Mod 10

If a > b And b < c And c > d Then

S = S + 1

End If

Next

[A1] = S

End Sub

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執行結果為 1695