Permutations and Combinations

The number of sequences of ten terms, whose terms are either $0$, $1$, or $2$, that contain exactly five $1$’s and exactly three $2$’s is equal to:

Line $L_1$ of slope $2$ and line $L_2$ of slope $\tfrac12$ intersect at the origin $O$. In the first quadrant, $P_1,P_2,\dots,P_{12}$ are 12 points on $L_1$ and $Q_1,Q_2,\dots,Q_9$ are 9 points on $L_2$. Then the total number of triangles that can be formed with vertices chosen from the 22 points $O,P_1,\dots,P_{12},Q_1,\dots,Q_9$ is:

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon $P$ of $n$ sides and $q$ be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p + q = 126$, then the eccentricity of the ellipse $\frac{x^2}{16} + \frac{y^2}{n} = 1$ is:

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is:

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is:

Let $S$ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $S$, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

Let $\mathrm{P}$ be the set of seven-digit numbers with sum of their digits equal to 11. If the numbers in $\mathrm{P}$ are formed by using the digits 1, 2, and 3 only, then the number of elements in the set $P$ is: