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

How many words can be formed by using all the letters of the word '$TRIVANDRUM$' using each letter exactly once (in $!$)?

$A$ person wants to climb a $n$-step staircase using one step or two steps. Let $C_n$ denote the number of ways of climbing the $n$-step staircase. Then $C_{18} + C_{19}$ equals

There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

In how many ways can the letters of the word '$NUTAN$' be arranged when consonants occupy odd places and vowels come on even places?

$A$ question paper is divided into two parts $A$ and $B$,and each part contains $5$ questions. The number of ways in which a candidate can answer $6$ questions by selecting at least two questions from each part is: