Let the mean and variance of 12 observations be $$\frac{9}{2}$$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $$\frac{m}{n}$$, where $$\mathrm{m}$$ and $$\mathrm{n}$$ are coprime, then $$\mathrm{m}+\mathrm{n}$$ is equal to :

The absolute difference of the coefficients of $$x^{10}$$ and $$x^{7}$$ in the expansion of $$\left(2 x^{2}+\frac{1}{2 x}\right)^{11}$$ is equal to :

The area of the quadrilateral $$\mathrm{ABCD}$$ with vertices $$\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$$ and $$\mathrm{D}(1,-6,-7)$$ is equal to :

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which $$\mathrm{C}$$ and $$\mathrm{S}$$ do not come together, is $$(6 !) \mathrm{k}$$, then $$\mathrm{k}$$ is equal to :