$a/ \ A^3_n+C^{n-2}_n=14n(n\ge3)\\ <=>\dfrac{n!}{(n-3)!}+\dfrac{n!}{2!(n-2)!}=14n\\ <=>(n-2)(n-1)n+\dfrac{n(n-1)}{2}=14n\\ <=>(n-1)\left(n-2+\dfrac{1}{2}\right)=14\\ <=>n^2-2,5n-12,5=0\\ <=>\left\{\begin{array}{l} n=5(TM)\\ n=-2,5(L) \end{array}\right.\\ b/ \ C^{n-2}_{n+1}+2C^3_{n-1}=7(n-1)(n\ge4)\\ <=>\dfrac{(n+1)!}{3!(n-2)!}+2\dfrac{(n-1)!}{3!(n-4)!}=7(n-1)\\ <=>\dfrac{(n-1)n(n+1)}{6}+\dfrac{(n-3)(n-2)(n-1)}{3}=7(n-1)\\ <=>n(n+1)+2(n-3)(n-2)=42\\ <=>3n^2-9n-30=0\\ <=>\left\{\begin{array}{l} n=5(TM)\\ n=-2(L) \end{array}\right.\\ c/ \ A^3_n -2C^4_n=3A^2_n(n\ge4)\\ <=>\dfrac{n!}{(n-3)!}-2\dfrac{n!}{4!(n-4)!}=3\dfrac{n!}{(n-2)!}\\ <=>(n-2)(n-1)n-2\dfrac{(n-3)(n-2)(n-1)n}{24}=3(n-1)n\\ <=>n-2-\dfrac{(n-2)(n-3)}{12}=3\\ <=>12n-24-n^2+5n-6=36\\ <=>n^2+17n-66\\ <=>\left\{\begin{array}{l} n=11\\ n=6 \end{array}\right.\\ d/ \ \dfrac{A^{n-3}_n}{n}=(n-2)!\\ <=>\dfrac{n!}{n(n-3)!}=(n-2)!\\ <=>1=(n-3)!\\ <=>n=4\\ e/ \ 3A^2_n+42=A^2_{2n}(n\ge2)\\ <=>3\dfrac{n!}{(n-2)!}+42=\dfrac{(2n)!}{(2n-2)!}\\ <=>3n(n-1)+42=2n(2n-1)\\ <=>-n^2-n+42=0\\ <=>\left\{\begin{array}{l} n=6(TM)\\ n=-7(L) \end{array}\right.\\ f/ \ 120A^{n-3}_{n-1}=A^n_{n+2}\\ <=>120\dfrac{(n-1)!}{2!}=\dfrac{(n+2)!}{2!}\\ <=>120=n(n-1)(n+2)\\ <=>n=4$
