If a is the coefficient of  $x^5$ in ${(1-x+x^2-x^3)}^5$,b is the coefficient of  $x^7$in ${(1-x-x^2+x^3)}^5$ and c is the coefficient of   in ${(1-x-x^2+x^3)}^6$, then the descending order  of a, b, c is 

${(1-x+x^2-x^3)}^5$

$={\left[(1-x)+x^2(1-x)\right]}^5$

$={(1-x)}^5{(1+x^2)}^5$

$$\begin{gathered} =({}^{\text{5}}{\text{C}}_{\text{0}}-{}^{\text{5}}{\text{C}}_{1}x+{}^{\text{5}}{\text{C}}_2{x}^2-{}^{\text{5}}{\text{C}}_3{x}^3+{}^{\text{5}}{\text{C}}_4{x}^4-{}^{\text{5}}{\text{C}}_5{x}^5) \\ \times ({}^{\text{5}}{\text{C}}_{\text{0}}+{}^{\text{5}}{\text{C}}_1{x}^{ }+{}^{\text{5}}{\text{C}}_2{x}^4+\cdots ) \end{gathered}$$

Given $a=x^5$ coefficient

$=-{}^{\text{5}}{\text{C}}_1.{}^{\text{5}}{\text{C}}_2-{}^{\text{5}}{\text{C}}_3.{}^{\text{5}}{\text{C}}_1-{}^{\text{5}}{\text{C}}_5.{}^{\text{5}}{\text{C}}_{\text{0}}$

$=-[5\times 10+5\times 10+1]$

$\begin{array}{l}=-101\\ {(1-x-x^2+x^3)}^5\\ {((1-x)-x^2(1-x))}^5\\ {(1-x)}^5{(1-x^2)}^5\\ {(1-x)}^{10}{(1+x)}^5\\ b=\text{coefficient\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}{x}^7\\ =[{}^{\text{10}}{\text{C}}_{\text{0}}-{}^{\text{10}}{\text{C}}_{1}x+{}^{\text{10}}{\text{C}}_2{x}^2+\cdots ]\\ [{}^5{\text{C}}_{\text{0}}+{}^5{\text{C}}_{1}x+{}^5{\text{C}}_2{x}^2+\cdots ]\end{array}$

$\begin{array}{l}={}^{\text{10}}{\text{C}}_2.{}^5{\text{C}}_5+(-{}^{\text{10}}{\text{C}}_3){}^5{\text{C}}_4+{}^{\text{10}}{\text{C}}_4{}^5{\text{C}}_3-{}^{\text{10}}{\text{C}}_5{}^5{\text{C}}_2\\ +{}^{\text{10}}{\text{C}}_6{}^5{\text{C}}_1-{}^{\text{10}}{\text{C}}_7.{}^5{\text{C}}_0\end{array}$

$=45-600+2100-2520+1050-120$

$=45$

$\begin{array}{l}{(1-x-x^2+x^3)}^6\\ {(1-x-x^2+(1-x))}^6\\ {(1-x)}^6+{(1-x^2)}^6\\ {(1-x)}^{12}+{(1+x)}^6\\ C=\text{coefficient.\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}}{x}^7\text{\hspace{0.17em}\hspace{0.17em}in\hspace{0.17em}\hspace{0.17em}}{(1-x-x^2+x^3)}^6={(1-x)}^{12}{(1+x)}^6\end{array}$

$\begin{array}{l}{(1-x)}^{12}{(1+x)}^6\\ =({}^{\text{12}}{\text{C}}_{\text{0}}-{}^{\text{12}}{\text{C}}_{1}x+{}^{\text{12}}{\text{C}}_2{x}^2+\cdots )({}^6{\text{C}}_{\text{0}}+{}^6{\text{C}}_{1}x+\cdots )\\ x^7\text{coefficient}=-{}^{\text{12}}{\text{C}}_1.{}^6{\text{C}}_6+{}^{\text{12}}{\text{C}}_2{}^6{\text{C}}_5-{}^{\text{12}}{\text{C}}_3.{}^6{\text{C}}_4+{}^{\text{12}}{\text{C}}_4{}^6{\text{C}}_3\\ -{}^{\text{12}}{\text{C}}_5{}^6{\text{C}}_2+{}^{\text{12}}{\text{C}}_6{}^6{\text{C}}_1-{}^{\text{12}}{\text{C}}_7{}^6{\text{C}}_{\text{0}}\end{array}$

$\begin{array}{l}C=(-12)+\left(66\right)\left(6\right)-3300+9900-11880+3128-792\\ =(396+9900+3128)-(12+3300++11880+792)\\ =13,424-15,984\\ C=-2560\end{array}$

Descending order is b, a, c