a)

• $\left(\begin{array}{c}6\\ 4\end{array}\right)=\frac{6!}{4!\cdot \left(6-4\right)!}=\frac{6!}{4!\cdot 2!}=\frac{\cancel{4!}\cdot 5\cdot 6}{\cancel{4!}\cdot 1\cdot 2}=15$  
• $\left(\begin{array}{c}9\\ 6\end{array}\right)=\frac{9!}{6!\cdot \left(9-6\right)!}=\frac{9!}{6!\cdot 3!}=\frac{\cancel{6!}\cdot 7\cdot 8\cdot 9}{\cancel{6!}\cdot 1\cdot 2\cdot 3}=84$  
• $\left(\begin{array}{c}13\\ 9\end{array}\right)=\frac{13!}{9!\cdot \left(13-9\right)!}=\frac{13!}{9!\cdot 4!}=\frac{\cancel{9!}\cdot 10\cdot 11\cdot 12\cdot 13}{\cancel{9!}\cdot 1\cdot 2\cdot 3\cdot 4}=715$  
• $\left(\begin{array}{c}2000\\ 1998\end{array}\right)=\frac{2000!}{1998!\cdot \left(2000-1998\right)!}=\frac{2000!}{1998!\cdot 2!}=\frac{\cancel{1998!}\cdot 1999\cdot 2000}{\cancel{1998!}\cdot 1\cdot 2}=1999000$  
• $\left(\begin{array}{c}25\\ 20\end{array}\right)=\frac{25!}{20!\cdot \left(25-20\right)!}=\frac{25!}{20!\cdot 5!}=\frac{\cancel{20!}\cdot 21\cdot 22\cdot 23\cdot 24\cdot 25}{\cancel{20!}\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5}=53130$  
• $\left(\begin{array}{c}49\\ 6\end{array}\right)=\frac{49!}{6!\cdot \left(49-6\right)!}=\frac{49!}{6!\cdot 43!}=\frac{\cancel{43!}\cdot 44\cdot 45\cdot 46\cdot 47\cdot 48\cdot 49}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot \cancel{43!}}=13983816$  

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b)

• $\left(\begin{array}{c}8\\ 8\end{array}\right)=\frac{8!}{8!\cdot \left(8-8\right)!}=\frac{8!}{8!\cdot 0!}=\frac{8!}{8!\cdot 1}=1$  
• $\left(\begin{array}{c}18\\ 17\end{array}\right)=\frac{18!}{17!\cdot \left(18-17\right)!}=\frac{18!}{17!\cdot 1!}=\frac{17!\cdot 18}{17!}=18$  
• $\left(\begin{array}{c}62\\ 1\end{array}\right)=\frac{62!}{1!\cdot \left(62-1\right)!}=\frac{62!}{61!}=\frac{61!\cdot 62}{61!}=62$  
• $\left(\begin{array}{c}17\\ 0\end{array}\right)=\frac{17!}{0!\cdot \left(17-0\right)!}=\frac{17!}{1\cdot 17!}=1$  
• $\left(\begin{array}{c}1234\\ 1233\end{array}\right)=\frac{1234!}{1233!\cdot \left(1234-1233\right)!}=\frac{1234!}{1233!\cdot 1!}=\frac{1233!\cdot 1234}{1233!}=1234$  
• $\left(\begin{array}{c}9876\\ 9876\end{array}\right)=\frac{9876!}{9876!\cdot \left(9876-9876\right)!}=\frac{9876!}{9876!\cdot 0!}=\frac{9876!}{9876!\cdot 1}=1$  

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c)

• $\frac{\left(\begin{array}{c}8\\ 6\end{array}\right)}{\left(\begin{array}{c}8\\ 2\end{array}\right)}=\frac{\frac{8!}{6!\cdot \left(8-6\right)!}}{\frac{8!}{2!\cdot \left(8-2\right)!}}=\frac{\frac{8!}{6!\cdot 2!}}{\frac{8!}{2!\cdot 6!}}=1$  
• $\frac{\left(\begin{array}{c}6\\ 4\end{array}\right)\cdot \left(\begin{array}{c}4\\ 2\end{array}\right)}{6!}=\frac{\frac{6!}{4!\cdot \left(6-4\right)!}\cdot \frac{4!}{2!\cdot \left(4-2\right)!}}{6!}=\frac{\frac{6!}{\cancel{4!}\cdot 2!}\cdot \frac{\cancel{4!}}{2!\cdot 2!}}{6!}=\frac{\frac{6!}{2\cdot 2\cdot 2}}{6!}=\frac{1}{8}$  
• $\frac{\left(\begin{array}{c}15\\ 10\end{array}\right)}{\left(15-10\right)!}=\frac{\frac{15!}{10!\cdot \left(15-10\right)!}}{\left(15-10\right)!}=\frac{\frac{15!}{10!\cdot 5!}}{5!}=\frac{15!}{10!\cdot 5!\cdot 5!}=\frac{10!\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15}{10!\cdot 120\cdot 120}=25,025$  
  
  
• $\frac{\left(\begin{array}{c}129\\ 101\end{array}\right)+\left(\begin{array}{c}129\\ 100\end{array}\right)}{\begin{array}{c}130\\ 101\end{array}}=\frac{\frac{129!}{101!\cdot \left(129-101\right)!}+\frac{129!}{100!\cdot \left(129-100\right)!}}{\frac{130!}{101!\cdot \left(130-101\right)!}}==\frac{\frac{129!}{101!\cdot 28!}+\frac{129!}{100!\cdot 29!}}{\frac{130!}{101!\cdot 29!}}=$  

$=\frac{\frac{129!}{100!\cdot 28!}\cdot \left(\frac{1}{101}+\frac{1}{29}\right)}{\frac{129!}{100!\cdot 28!}\cdot \frac{130}{101\cdot 29}}=\frac{\frac{29+101}{101\cdot 29}}{\frac{130}{101\cdot 29}}=\frac{\frac{130}{101\cdot 29}}{\frac{130}{101\cdot 29}}=1$