In how many ivays can 20 identical toys be distributed among 4 children so that each one gets at least 3 toys?

Required number of ways

=Coefficient of $x^{20}\text{ in }{\left(x^3+x^4+\dots +x^{20}\right)}^4$

= Coefficient of $x^8\text{ in }{\left(1+x+\dots +x^{17}\right)}^4$

= Coefficient of $x^8\mathrm{in}{\left(\frac{1-x^{18}}{1-x}\right)}^4$

= Coefficient of $x^8\text{ in }(1-x{)}^{-4}$

${=}^{8+4-1}{C}_{4-1}\left[\because \text{ Coeff. of }{x}^n\text{ in }(1-x{)}^{-r}{=}^{n+r-1}{C}_{r-1}\right]$

${=}^{11}{C}_3$