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Sum of product of binomial coefficients

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Question

Expand the expression $(2 x - 3)^{6}$

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Solution

Using Binomial theorem,
$(2 x - 3)^{6} =^{6} C_{0} (2 x)^{6} -^{6} C_{1} (2 x)^{5} (3) +^{6} C_{2} (2 x)^{4} (3)^{2} -^{6} C_{3} (2 x)^{3} (3)^{3} +^{6} C_{4} (2 x)^{2} (3)^{4} -^{6} C_{5} (2 x) (3)^{5} +^{6} C_{6} (3)^{6}$
$= 64 x^{6} - 6 (32 x^{5}) (3) + 15 (16 x^{4}) (9) - 20 (8 x^{3}) (27) + 15 (4 x^{2}) (81) - 6 (2 x) (243) + 729$
$= 64 x^{6} - 576 x^{5} + 2160 x^{4} - 4320 x^{3} + 4860 x^{2} - 2916 x + 729$

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NCERT - Standard XI

Sum of product of binomial coefficients

Standard XI Mathematics

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