If ${ }^n \mathrm{C}_0+\frac{1}{2}{ }^n \mathrm{C}_1+\frac{1}{3}{ }^n \mathrm{C}_2$$$+\ldots \frac{1}{n}^n C_{n-1}+\frac{1}{n+1}{ }^n C_n=\frac{1023}{10} \,\,\, then \,\,\,\,\mathrm{n}=$$

A pair of fair dice is thrown 4 times. If getting the same number on both dice is considered as a success, then the probability of two successes are

The position vectors of the points $A, B, C$ are $\hat{i}+2 \hat{j}-\hat{k}, \hat{i}+\hat{j}+\hat{k}, 2 \hat{i}+3 \hat{j}+2 \hat{k}$ respectively. If $A$ is chosen as the origin, then the cross product of position vectors of $B$ and $C$ are

If the area of a parallelogram whose diagonals are represented by vectors $3 \hat{i}+\lambda \hat{j}+2 \hat{k}$ and $\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ is $\frac{\sqrt{117}}{2}$ sq. units, then $\lambda=$