Question

. Determine the value of each binomial coefficient.

a. $\left(\begin{array}{l}8\\ 3\end{array}\right)$

b. $\left(\begin{array}{l}8\\ 5\end{array}\right)$

c. $\left(\begin{array}{l}6\\ 6\end{array}\right)$

d. $\left(\begin{array}{l}10\\ 2\end{array}\right)$

e. $\left(\begin{array}{c}40\\ 4\end{array}\right)$

f. $\left(\begin{array}{c}100\\ 0\end{array}\right)$

Short answer

Part a)$56$

Part b)$56$

Part c)$1$

Part d)$45$

Part e)$91390$

Part f)$1$

Step-by-step solution

Step 1:Given Information

a.$\left(\begin{array}{l}8\\ 3\end{array}\right)$

b. $\left(\begin{array}{l}8\\ 5\end{array}\right)$

c. $\left(\begin{array}{l}6\\ 6\end{array}\right)$

d.$\left(\begin{array}{c}10\\ 2\end{array}\right)$

e.$\left(\begin{array}{c}40\\ 4\end{array}\right)$

f.$\left(\begin{array}{c}100\\ 0\end{array}\right)$

Step 2:Simplification Part a)

$\left(\begin{array}{l}8\\ 3\end{array}\right)=\frac{8!}{3!\times (8-3)!}$

$=\frac{8!}{3!\times 5!}$

$=56$

Step 2:Simplification Part b)

$\left(\begin{array}{l}8\\ 5\end{array}\right)=\frac{8!}{5!\times (8-5)!}$

$=\frac{8!}{3!\times 5!}$

$=56$

Step 2:Simplification Part c)

$\left(\begin{array}{l}6\\ 6\end{array}\right)=\frac{6!}{6!\times (6-6)!}$

$=\frac{6!}{6!\times 0!}$

$=1$

Step 2:Simplification Part d)

$\left(\begin{array}{c}10\\ 2\end{array}\right)=\frac{10!}{2!\times (10-2)!}$

$=\frac{6!}{2!\times 8!}$

$=45$

Step 2:Simplification Part e)

$\left(\begin{array}{c}40\\ 4\end{array}\right)=\frac{40!}{4!\times (40-4)!}$

$=\frac{40!}{4!\times 36!}$

$=91390$

Step 2:Simplification Part f)

$\left(\begin{array}{c}100\\ 0\end{array}\right)=\frac{100!}{0!\times (100-0)!}$

$=\frac{100!}{0!\times 100!}$

$=1$