a) 

$2!=2\cdot 1=2$  

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$4!=4\cdot 3\cdot 2\cdot 1=24$  

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$7!=7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=5040$  

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$11!=11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=39916800$  

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b) 

$2!+0!=2\cdot 1+1=2+1=3$  

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$2!+2!=2\cdot 1+2\cdot 1=2+2=4$  

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$1!+6!=1+6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=1+720=721$  

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$13!-2!=13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1-2\cdot 1=6227020800-2=6227020798$  

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c) 

$\frac{5!}{3!}=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1}=\frac{5\cdot 4\cdot \cancel{3!}}{\cancel{3!}}=5\cdot 4=20$  

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$\frac{10!}{8!}=\frac{10\cdot 9\cdot \cancel{8!}}{\cancel{8!}}=10\cdot 9=90$  

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$\frac{21!}{8!}=\frac{21\cdot 20\cdot 19\cdot 18\cdot \cancel{17!}}{\cancel{17!}}=21\cdot 20\cdot 19\cdot 18=143640$  

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$\frac{18!}{8!}=\frac{18\cdot 17\cdot 16\cdot 15\cdot 14\cdot 13\cdot \cancel{12!}}{\cancel{12!}}=18\cdot 17\cdot 16\cdot 15\cdot 14\cdot 13=13366080$  

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d) 

$\frac{13!}{10!\cdot 6!}=\frac{13\cdot 12\cdot 11\cdot \cancel{10!}}{\cancel{10!}\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}=\frac{13\cdot \cancel{12}\cdot 11}{\cancel{6}\cdot 5\cdot 4\cdot 3\cdot \cancel{2}\cdot 1}=\frac{13\cdot 11}{5\cdot 4\cdot 3}=\frac{143}{60}$  

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$\frac{4!\cdot 15!}{16!}=\frac{4\cdot 3\cdot 2\cdot 1\cdot \cancel{15!}}{16\cdot \cancel{15!}}=\frac{\cancel{4}\cdot 3\cdot \cancel{2}\cdot 1}{{\cancel{16}}_2}=\frac{3}{2}$  

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$\frac{6!\left(3!+5!\right)}{7!-3!}=\frac{6!\left(3!+5\cdot 4\cdot 3!\right)}{7\cdot 6\cdot 5\cdot 4\cdot 3!-3!}=\frac{6!\left(\cancel{3!}\left(1+5\cdot 4\right)\right)}{\cancel{3!}\left(7\cdot 6\cdot 5\cdot 4-1\right)}=\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\left(1+20\right)}{840-1}=\frac{720\cdot 21}{839}=\frac{15120}{839}$  

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$5!\left(\frac{8!}{4!}+\frac{7!}{5!}\right)=5!\left(\frac{8\cdot 7\cdot 6\cdot 5\cdot \cancel{4!}}{\cancel{4!}}+\frac{7\cdot 6\cdot \cancel{5!}}{\cancel{5!}}\right)=5!\left(8\cdot 7\cdot 6\cdot 5+7\cdot 6\right)=5\cdot 4\cdot 3\cdot 2\cdot 1\left(1680+42\right)=120\cdot 1722=206640$  

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e) 

$\frac{\left(n+1\right)!}{n!}=\frac{\left(n+1\right)\cancel{n!}}{\cancel{n!}}=n+1$  

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$\frac{\left(n+2\right)!}{n!}=\frac{\left(n+2\right)\left(n+1\right)\cancel{n!}}{\cancel{n!}}=\left(n+2\right)\left(n+1\right)=n^2+n+2n+2=n^2+3n+2$  

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$\frac{\left(n+1\right)!}{\left(n-1\right)!\cdot n}=\frac{\left(n+1\right)!}{n!}=\frac{\left(n+1\right)\cancel{n!}}{\cancel{n!}}=n+1$  

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$\frac{\left(n+2\right)!-n!}{n!-\left(n-1\right)!}=\frac{\left(n+2\right)\left(n+1\right)n!-n!}{n\left(n-1\right)!-\left(n-1\right)!}=\frac{n!\left[\left(n+2\right)\left(n+1\right)-1\right]}{\left(n-1\right)!\left(n-1\right)}=\frac{n\cancel{\left(n-1\right)!}\left(n^2+n+2n+2-1\right)}{\cancel{\left(n-1\right)!}\left(n-1\right)}=\frac{n\left(n^2+3n+1\right)}{n-1}=\frac{n^3+3n^2+1n}{n-1}$