Question

Find the value of \(\left( {\begin{array}{*{20}{c}}{\frac{3}{2}}\\4\end{array}} \right)\).

Short answer

The value of \(\left( {\begin{array}{{}{}}{\frac{3}{2}}\\4\end{array}} \right)\) is \(\frac{3}{{128}}\).

Step-by-step solution

Given information

Referring to the equations 5.3.13 and 5.3.14.

If m and r both are positive integers, then the binomial coefficient is given as,\(\begin{array}{c}\left( {\begin{array}{{}{}}m\\r\end{array}} \right) = \frac{{m!}}{{r!\left( {m - r} \right)!}}\\ = \frac{{m\left( {m - 1} \right)...\left( {m - r + 1} \right)}}{{r!}}\end{array}\)

Compute the required value

The value of \(\left( {\begin{array}{{}{}}{\frac{3}{2}}\\4\end{array}} \right)\) is given by,

\(\begin{array}{}\left( {\begin{array}{{}{}}{\frac{3}{2}}\\4\end{array}} \right) = \frac{{\left( {\frac{3}{2}} \right)!}}{{4!\left( {\frac{3}{2} - 4} \right)!}}\\ = \frac{{\frac{3}{2}\left( {\frac{3}{2} - 1} \right)\left( {\frac{3}{2} - 2} \right)\left( {\frac{3}{2} - 4 + 1} \right)}}{{4!}}\\ = \frac{{\frac{3}{2} \times \frac{1}{2} \times \left( { - \frac{1}{2}} \right) \times \left( { - \frac{3}{2}} \right)}}{{4 \times 3 \times 2 \times 1}}\\ = \frac{3}{{128}}\end{array}\)

Therefore, the value of \(\left( {\begin{array}{{}{}}{\frac{3}{2}}\\4\end{array}} \right)\) is \(\frac{3}{{128}}\).