There are $10$ points in a plane,of which no three points are collinear except $4$. Then,the number of distinct triangles that can be formed by joining any three points of these ten points,such that at least one of the vertices of every triangle formed is from the given $4$ collinear points is

The number of ways in which the letters $A, B, C, D, E$ can be placed in the $8$ boxes of the figure below so that no row remains empty and at most one letter can be placed in a box is:

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle,then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

Given six line segments of lengths $2, 3, 4, 5, 6, 7$ units,the number of triangles that can be formed by these lines is

The number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):

If in a regular polygon the number of diagonals is $54$,then the number of sides of this polygon is