If `alpha=\ \ ^m C_2,\ ` then find the value of `\ ^(alpha)C_2dot`

Explanation

`\ ^{alpha} C_{2}=frac{alpha}{2} times frac{(alpha-1)}{1} times\ ^{alpha} C_{0}[because^{n} C_{r}=frac{n}{r} cdot\ ^{n-1} C_{r-1}]`

`\ ^{n} C_{r}=frac{n}{r} cdot\ ^{n-1} C_{r-1}[because^{n} C_{0}=1]`

`=frac{1}{2}[\ ^{m} C_{2}(\ ^{m} C_{2}-1)]`

`=frac{1}{2}[frac{m !}{2 !(m-2) !}(frac{m !}{2 !(m-2) !}-1)]`

`=frac{1}{2}[frac{m(m-1)}{2}(frac{m(m-1)}{2}-1)]`

`=frac{1}{2}[frac{m(m-1)}{2}(frac{m(m-1)-2}{2})]`

`=frac{1}{8}[m(m-1){m(m-1)-2}]`

`=frac{1}{8}[m^{2}(m-1)^{2}-2 m(m-1)]`

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