Tính: a) S = C 0 2022 9^2022 + C 1 2022 9^2021 + + C k 2022 9^(2022 - k) + + C 2021 2022 9 + C 2022 2022

Áp dụng công thức nhị thức Newton ta có:

a) $S=C_{{}_{2022}}^0{9}^{2022}+C_{{}_{2022}}^1{9}^{2021}+\mathrm{...}+C_{{}_{2022}}^k{9}^{2022-k}+\mathrm{...}+C_{{}_{2022}}^{2021}9+C_{2022}^{2022}$

$=C_{{}_{2022}}^0{9}^{2022}+C_{{}_{2022}}^1{9}^{2021}.1+\mathrm{...}+C_{{}_{2022}}^k{9}^{2022-k}{.1}^k+\mathrm{...}+C_{{}_{2022}}^{2021}{9.1}^{2021}+C_{2022}^{2022}{.1}^{2022}$

$={\left(9+1\right)}^{2020}={10}^{2022}.$

b) $T=C_{2022}^0{4}^{2022}-C_{2022}^1{4}^{2021}.3+\mathrm{...}-C_{2022}^{2021}4.3^{2021}+C_{2022}^{2022}{3}^{2022}$

$=C_{2022}^0{4}^{2022}+C_{2022}^1{4}^{2021}.{\left(-3\right)}^1+\mathrm{...}+C_{2022}^{2021}4.{\left(-3\right)}^{2021}+C_{2022}^{2022}{\left(-3\right)}^{2022}$

$={\left[4+\left(-3\right)\right]}^{2022}=1^{2022}=1.$