Question Number 64702 by Tanmay chaudhury last updated on 20/Jul/19
Answered by Tanmay chaudhury last updated on 20/Jul/19
$${abstaining}={a}\:\:\:\:{voting}={v}\:\: \\ $$$${a}=\mathrm{1}+{v}=\mathrm{7} \\ $$$${a}=\mathrm{2}+{v}=\mathrm{6} \\ $$$$... \\ $$$${a}=\mathrm{0}+{v}=\mathrm{8} \\ $$$$\left({a}+{v}\right)^{\mathrm{8}} ={a}^{\mathrm{8}} +\mathrm{8}{c}_{\mathrm{1}} {a}^{\mathrm{7}} {v}+\mathrm{8}{c}_{\mathrm{2}} {a}^{\mathrm{6}} {v}^{\mathrm{2}} +...+{v}^{\mathrm{8}} \\ $$$${look}\:{red}\:{marked}\:{a}^{\mathrm{8}} \:{not}\:{allowd}\: \\ $$$${all}\:\mathrm{8}\:{issues}\:\:{con}\:{not}\:{be}\:{kept}\:{abstained} \\ $$$$\boldsymbol{{voting}}\:\mathrm{8}\:\boldsymbol{{issues}} \\ $$$$\left(\boldsymbol{{y}}=\mathrm{0}+\boldsymbol{{n}}=\mathrm{8}\right),\left({y}=\mathrm{1}+{n}=\mathrm{7}\right)...\left({y}=\mathrm{8}+{n}=\mathrm{0}\right) \\ $$$$\left({n}+{y}\right)^{\mathrm{8}} ={n}^{\mathrm{8}} +\mathrm{8}{c}_{\mathrm{1}} {n}^{\mathrm{7}} {y}^{\mathrm{1}} +..+{y}^{\mathrm{8}} =\mathrm{2}^{\mathrm{8}} =\mathrm{256} \\ $$$${voting}\:\mathrm{7}\:{issues}\left(\boldsymbol{{y}}=\boldsymbol{{yes}}\:\:\boldsymbol{{n}}=\boldsymbol{{no}}\right) \\ $$$$\left({y}=\mathrm{0}+{n}=\mathrm{7}\right),\left({y}=\mathrm{1}+{n}=\mathrm{6}\right)...\left({y}=\mathrm{7}+{n}=\mathrm{0}\right) \\ $$$$\left({n}+{y}\right)^{\mathrm{7}} ={n}^{\mathrm{7}} +\mathrm{7}{c}_{\mathrm{1}} {n}^{\mathrm{6}} {y}^{\mathrm{1}} +\mathrm{7}{c}_{\mathrm{2}} {n}^{\mathrm{5}} {y}^{\mathrm{2}} +..+{y}^{\mathrm{7}} =\mathrm{2}^{\mathrm{7}} =\mathrm{128} \\ $$$$\boldsymbol{{votin}}\:\mathrm{6}\:\boldsymbol{{issues}}= \\ $$$$\left(\boldsymbol{{n}}+\boldsymbol{{y}}\right)^{\mathrm{6}} =\boldsymbol{{n}}^{\mathrm{6}} +\mathrm{6}\boldsymbol{{c}}_{\mathrm{1}} \boldsymbol{{n}}^{\mathrm{5}} \boldsymbol{{y}}+...+\boldsymbol{{y}}^{\mathrm{6}} =\mathrm{2}^{\mathrm{6}} =\mathrm{64}\:\:\left(\:\:\:\boldsymbol{{not}}\:\boldsymbol{{completed}}\:\right) \\ $$$$..{i}\:{am}\:{getting}... \\ $$$$=\mathrm{8}\boldsymbol{{c}}_{\mathrm{1}} ×\mathrm{2}=\mathrm{16} \\ $$$$\:\mathrm{8}\boldsymbol{{c}}_{\mathrm{2}} ×\mathrm{2}^{\mathrm{2}} =\mathrm{28}×\mathrm{4}=\mathrm{112} \\ $$$$\mathrm{8}\boldsymbol{{c}}_{\mathrm{3}} ×\mathrm{2}^{\mathrm{3}} =\frac{\mathrm{8}×\mathrm{7}×\mathrm{6}}{\mathrm{3}×\mathrm{2}}×\mathrm{8}=\mathrm{448}\:\:\: \\ $$$$\mathrm{8}\boldsymbol{{c}}_{\mathrm{4}} ×\mathrm{2}^{\mathrm{4}} =\frac{\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}}{\mathrm{4}×\mathrm{3}×\mathrm{2}}×\mathrm{16}=\mathrm{1120} \\ $$$$\mathrm{8}{c}_{\mathrm{5}} ×\mathrm{2}^{\mathrm{5}} =\frac{\mathrm{8}×\mathrm{7}×\mathrm{6}}{\mathrm{3}×\mathrm{2}}×\mathrm{32}=\mathrm{1792} \\ $$$$\mathrm{8}{c}_{\mathrm{6}} ×\mathrm{2}^{\mathrm{6}} =\mathrm{28}×\mathrm{64}=\mathrm{1792} \\ $$$$\mathrm{8}{c}_{\mathrm{7}} ×\mathrm{2}^{\mathrm{7}} =\mathrm{8}×\mathrm{128}=\mathrm{1024} \\ $$$$\boldsymbol{{i}}{o}\:\boldsymbol{{am}}\:\boldsymbol{{getting}}=\mathrm{6304}+\mathrm{256}=\mathrm{6560}\: \\ $$$$\mathrm{256}\:{is}\:{when}\:{all}\:{issues}\:{are}\:{attended}\:{no}\:{abstaining} \\ $$$$\boldsymbol{{final}}\:\boldsymbol{{answer}}=\mathrm{6560} \\ $$$$ \\ $$