# 點算的奧秘：列舉型母函數

(1 + x)3 = (1 + x) × (1 + x) × (1 + x)

 (1 + ax) × (1 + bx) × (1 + cx) = 1 + (a + b + c)x + (ab + bc + ac)x2 + abcx3     (1)

(1 + x)3 = 1 + 3x + 3x2 + x3    (2)

Σ0 ≤ r < ∞ fr(a1, ... an) xr    (3)

Σ0 ≤ r < ∞ fr(1, ... 1) xr    (4)

 [1 + Ax] × [Bx + (Bx)2 + (Bx)3 + ...] × [Cx + (Cx)2] = BCx2 × [1 + Ax] × [1 + Bx + (Bx)2 + ...] × [1 + Cx]

B5C2 + B6C + AB4C2 + AB5C

C(n, r) = P(n, r) / r!

 (A + B)3 = (A + B)(A + B)(A + B) = AAA + AAB + ABA + ABB + BAA + BAB + BBA + BBB    (5)

(A1 + ... An)r

Σ0 ≤ r < ∞ (A1 + ... An)r xr    (6)

(Aa + Bb)3

 (Aa + Bb)3 = AAAa3 + (AAB + ABA + BAA)a2b + (ABB + BAB + BBA)ab2 + BBBb3

(A1a1 + ... Anan)r     (7)

[1 + Ax + (Ax)2 + ...] × [Bx + (Bx)3 + (Bx)5 + ...] × [1 + (Cx)2 + (Cx)4 + ...]

ABC2 + AB3 + A3B

(Aa + Bb + Cc)4 = (Aa + Bb + Cc)(Aa + Bb + Cc)(Aa + Bb + Cc)(Aa + Bb + Cc)

ABCC + ACBC + ACCB + BACC + BCAC + BCCA + CABC + CACB + CBAC + CBCA + CCAB + CCBA

ABBB + BABB + BBAB + BBBA

AAAB + AABA + ABAA + BAAA

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