Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is

In how many different ways can the letters of the word '$AUCTION$' be arranged in such a way that the vowels always come together?

The number of groups that can be made from $5$ different green balls,$4$ different blue balls,and $3$ different red balls,if at least $1$ green and $1$ blue ball is to be included.

The number of ordered pairs $(r, k)$ for which $6 \cdot ^{35} C_{r} = (k^2 - 3) \cdot ^{36} C_{r+1}$,where $k$ is an integer,is:

The number of $4$-digit even numbers that can be formed using $0, 1, 2, 3, 4, 5, 6$ without repetition is

The number of triangles that can be formed by choosing the vertices from a set of $12$ points,seven of which lie on the same straight line,is