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## Q. 10.5.1

Evaluating Binomial Coefficients

Evaluate:

a. $\left(\begin{array}{l}6 \\2\end{array}\right)$         b. $\left(\begin{array}{l}3 \\0\end{array}\right)$          c. $\left(\begin{array}{l}9 \\3\end{array}\right)$          d. $\left(\begin{array}{l}4 \\4\end{array}\right)$.

## Verified Solution

In each case, we apply the definition of the binomial coefficient.

a. $\left(\begin{array}{l}6 \\2\end{array}\right)=\frac{6 !}{2 !(6-2) !}=\frac{6 !}{2 ! 4 !}=\frac{6 \cdot 5 \cdot \cancel{4!}}{2 \cdot 1 \cdot \cancel{4!}}=15$

c. $\left(\begin{array}{l}9 \\3\end{array}\right)=\frac{9 !}{3 !(9-3) !}=\frac{9 !}{3 ! 6 !}=\frac{9 \cdot 8 \cdot 7 \cdot \cancel{6 !}}{3 \cdot 2 \cdot 1 \cdot \cancel{6 !}}=84$

d. $\left(\begin{array}{l}4 \\4\end{array}\right)=\frac{4 !}{4 !(4-4) !}=\frac{\cancel{4 !}}{\cancel{4 !} 0 !}=\frac{1}{1}=1$