Question Number 63835 by gunawan last updated on 10/Jul/19
$$\mathrm{If}\:{n}\:\in\:{N},\:\mathrm{then}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\left(\mathrm{5}{x}−\mathrm{4}{y}\right)^{{n}} \:\mathrm{is} \\ $$
Commented by mr W last updated on 10/Jul/19
$$\left(\mathrm{5}{x}−\mathrm{4}{y}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{k}} {x}^{{k}} {y}^{{n}−{k}} \\ $$$$\left(\mathrm{5}×\mathrm{1}−\mathrm{4}×\mathrm{1}\right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{k}} \\ $$$$\Rightarrow{sum}\:{of}\:{coef}.=\left(\mathrm{5}×\mathrm{1}−\mathrm{4}×\mathrm{1}\right)^{{n}} =\mathrm{1}^{{n}} =\mathrm{1} \\ $$