Question:
if c0 c1x c2xsq ........... cn x to the power n=(1 x) hole to the power n . then proof that c0 sq c1 sq c2 sq ......... cn sq = (2n)!
Answer:
Given, (1 + x)n = C0 + C1 x + C2 x2 + ..............+ Cn xn ..........1
and (1 + x)n = C0 xn + C1 xn-1 + C2 xn-2 + ..............Cr xn-r + ..........+ Cn-1 x + Cn ...........2
Multiply 1 and 2, we get
(1 + x)2n = (C0 + C1 x + C2 x2 + ..............+ Cn xn ) * (C0 xn + C1 xn-1 + C2 xn-2 + ..............Cr xn-r + ..........+ Cn-1 x + Cn )
Now, equating the coefficient of xn on both side, we get
C02 + C12 + C22 + ..............+ Cnn = 2nCn = (2n)!/(n! * n!)
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