Let Sn=nC0nC3+nC1nC4+nC2nC5+ +nCn-3nCn(1+x)n=nC0+nC1x+nC2x2+nC3x3+nC0x4+ +nCnxn1+1xn=nC0+nC11x+nC21x2+nC31x3+ +nCn1xnSn=Coefficient of x3 in (1+x)n1+1xn=Coefficient of x3 in 1+x2nxn=Coefficient of xn+1 in 1+x2n=2nCn+1

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Let S_{n} =^{n} {C_{0}}^{n} C_{3} +^{n} {C_{1}}^{n} C_{4} +^{n} {C_{2}}^{n} C_{5} + . . +^{n} {C_{n - 3}}^{n} C_{n} (1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{3} x^{3} +^{n} C_{0} x^{4} + . . . +^{n} C_{n} x^{n} {(1 + \frac{1}{x})}^{n} =^{n} C_{0} +^{n} C_{1} (\frac{1}{x}) +^{n} C_{2} {(\frac{1}{x})}^{2} +^{n} C_{3} {(\frac{1}{x})}^{3} + . . . +^{n} C_{n} {(\frac{1}{x})}^{n} S_{n} = Coefficient of x^{3} in (1 + x)^{n} {(1 + \frac{1}{x})}^{n} = Coefficient of x^{3} in \frac{{(1 + x)}^{2 n}}{x^{n}} = Coefficient of x^{n + 1} in {(1 + x)}^{2 n} =^{2 n} C_{n + 1}

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