If $12 \cdot {}^{n}C_{2} = {}^{2n}C_{3}$,find $n$.

There are $3$ sections in a question paper and each section contains $5$ questions. $A$ candidate has to answer a total of $5$ questions,choosing at least one question from each section. Then the number of ways,in which the candidate can choose the questions,is

How many numbers lying between $99$ and $1000$ can be formed using the digits $2, 3, 7, 0, 8, 6$,if each digit can be used only once in each number?

$P_1$ and $P_2$ are two distinct and intersecting planes. Three non-collinear points lie on $P_1$ and another three non-collinear points lie on $P_2$ (none being on the line of intersection of the planes). Then,the maximum number of tetrahedrons formed using these six points is:

How many lines can be drawn through $21$ points on a circle?

In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ courses are compulsory for every student?