Evaluate :

To find: Value of $(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}$

Formula used: (i) ${ }^{n} C_{r}=\frac{n !}{(n-r) !(r) !}$

(ii) $(a+b)^{n}={ }^{n} C_{0} a^{n}+{ }^{n} C_{1} a^{n-1} b+{ }^{n} C_{2} a^{n-2} b^{2}+\ldots \ldots+{ }^{n} C_{n-1} a b^{n-1}+{ }^{n} C_{n} b^{n}$

$(a+b)^{6}={ }^{6} C_{0} a^{6}+{ }^{6} C_{1} a^{6-1} b+{ }^{6} C_{2} a^{6-2} b^{2}+{ }^{6} C_{3} a^{6-3} b^{3}+{ }^{6} C_{4} a^{6-4} b^{4}+{ }^{6} C_{5} a^{6-5} b^{5}+{ }^{6} C_{6} b^{6}$

$\Rightarrow{ }^{6} \mathrm{C} 0 \mathrm{a}^{6}+{ }^{6} \mathrm{C} 1 \mathrm{a}^{5} \mathrm{~b}+{ }^{6} \mathrm{C} 2 \mathrm{a}^{4} \mathrm{~b}^{2}+{ }^{6} \mathrm{C} 3 \mathrm{a}^{3} \mathrm{~b}^{3}+{ }^{6} \mathrm{C} 4 \mathrm{a}^{2} \mathrm{~b}^{4}+{ }^{6} \mathrm{C} 5 \mathrm{ab}^{5}+{ }^{6} \mathrm{C} 6 \mathrm{~b}^{6} \ldots$ (i)

$(a-b)^{6}=$

$=\left[{ }^{6} C_{0} a^{6}\right]+\left[{ }^{6} C_{1} a^{6-1}(-b)\right]+\left[{ }^{6} C_{2} a^{6-2}(-b)^{2}\right]+\left[{ }^{6} C_{3} a^{6-3}(-b)^{3}\right]+$

$\left[{ }^{6} C_{4} a^{6-4}(-b)^{4}\right]+\left[{ }^{6} C_{5} a^{6-5}(-b)^{5}\right]+\left[{ }^{6} C_{6}(-b)^{6}\right]$

$\Rightarrow{ }^{6} \mathrm{C}_{0} \mathrm{a}^{6}-{ }^{6} \mathrm{C}_{1} \mathrm{a}^{5} \mathrm{~b}+{ }^{6} \mathrm{C}_{2} \mathrm{a}^{4} \mathrm{~b}^{2}-{ }^{6} \mathrm{C}_{3} \mathrm{a}^{3} \mathrm{~b}^{3}+{ }^{6} \mathrm{C}_{4} \mathrm{a}^{2} \mathrm{~b}^{4}-{ }^{6} \mathrm{C}_{5} \mathrm{ab}^{5}+{ }^{6} \mathrm{C}_{6} \mathrm{~b}^{6} \ldots$ (ii)

Substracting (ii) from (i)

$(a+b)^{6}-(a-b)^{6}=\left[{ }^{6} C_{0} a^{6}+{ }^{6} C_{1} a^{5} b+{ }^{6} C_{2} a^{4} b^{2}+{ }^{6} C_{3} a^{3} b^{3}+{ }^{6} C_{4} a^{2} b^{4}+{ }^{6} C_{5} a b^{5}+{ }^{6} C_{6} b^{6}\right]-$

$\left[{ }^{6} C_{0} a^{6}-{ }^{6} C_{1} a^{5} b+{ }^{6} C_{2} a^{4} b^{2}-{ }^{6} C_{3} a^{3} b^{3}+{ }^{6} C_{4} a^{2} b^{4}-{ }^{6} C_{5} a b^{5}+{ }^{6} C_{6} b^{6}\right]$

$=2\left[{ }^{6} \mathrm{C}_{1} \mathrm{a}^{5} \mathrm{~b}+{ }^{6} \mathrm{C} 3 \mathrm{a}^{3} \mathrm{~b}^{3}+{ }^{6} \mathrm{C} 5 \mathrm{ab}^{5}\right]$

$=2^{\left[\left\{\frac{6 !}{1 !(6-1) !} a^{5} a\right\}+\left\{\frac{6 !}{3 !(6-3) !} a^{3} b^{3}\right\}+\left\{\frac{6 !}{5 !(6-5) !} a b^{5}\right\}\right]}$

$=2\left[(6) a^{5} b+(20) a^{3} b^{3}+(6) a b^{5}\right]$

$\Rightarrow(a+b)^{6}-(a-b)^{6}=2\left[(6) a^{5} b+(20) a^{3} b^{3}+(6) a b^{5}\right]$

Putting the value of $a=\sqrt{3}$ and $b=\sqrt{2}$ in the above equation

$(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}$

$\Rightarrow 2\left[(6)(\sqrt{3})^{5}(\sqrt{2})+(20)(\sqrt{3})^{3}(\sqrt{2})^{3}+(6)(\sqrt{3})(\sqrt{2})^{5}\right]$

$\Rightarrow 2^{[54(\sqrt{6})+120(\sqrt{6})+24(\sqrt{6})]}$

$\Rightarrow 396 \sqrt{6}$

Ans) $396 \sqrt{6}$