The sum of the series 2 x 1 x 20C4 - 3 x 2 x 20C5 + 4 x 3 x 20C6 - 5 x 4 x 20C7 + ... + 18 x 17 x 20C20, is equal to

Question

The sum of the series \(2 \times 1 \times{ }^{20} \mathrm{C}_{4}-3 \times 2 \times{ }^{20} \mathrm{C}_{5}+4 \times 3 \times{ }^{20} \mathrm{C}_{6}-5 \times 4 \times { }^{20} \mathrm{C}_{7}+\ldots+18 \times 17 \times{ }^{20} \mathrm{C}_{20}, \) is equal to 

Answer 1

Answer is: 34 

\((1-\mathrm{x})^{20}={ }^{20} \mathrm{C}_{0}-{ }^{20} \mathrm{C}_{1} \mathrm{x}+{ }^{20} \mathrm{C}_{2} \mathrm{x}^{2} \ldots . .+{ }^{20} \mathrm{C}_{20} \mathrm{x}^{20}\)

\(\frac{(1-x)^{20}}{x^{2}}=\frac{{ }^{20} C_{0}}{\mathrm{x}^{2}}-\frac{{ }^{20} \mathrm{C}_{1}}{\mathrm{x}}+{ }^{20} \mathrm{C}_{2}-{ }^{20} \mathrm{C}_{3} \mathrm{x}+{ }^{20} \mathrm{C}_{4} \mathrm{x}^{2} \ldots.\)

Diff twice and put \(\mathrm{x}=1\)

\(=6-{ }^{20} \mathrm{C}_{1}(2)+\mathrm{A}\)

\(\mathrm{A}=40-6=34\)