The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by

The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$ is

Let ${}^nC_{r-1}=28$,${}^nC_r=56$,and ${}^nC_{r+1}=70$. Let $A(4 \cos t, 4 \sin t)$,$B(2 \sin t, -2 \cos t)$,and $C(3r - n, r^2 - n - 1)$ be the vertices of a triangle $ABC$,where $t$ is a parameter. If $(3x - 1)^2 + (3y)^2 = \alpha$ is the locus of the centroid of triangle $ABC$,then $\alpha$ equals:

The number of integers $n$ such that $100 \leq n \leq 999$ and $n$ contains at most two distinct digits is:

An envelope has space for at most $3$ stamps. If you are given three stamps of denomination $1$ and three stamps of denomination $a$ (where $a > 1$),what is the least positive integer for which there is no possible stamp value?

The remainder obtained when $1! + 2! + 3! + \ldots + 11!$ is divided by $12$ is