The sum of all the $4$-digit numbers formed by taking all the digits from ${0, 3, 6, 9}$ without repetition 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 positive integers less than $10000$ which contain the digit $5$ at least once is

Let $0 \leq r \leq n$. If ${ }^{n+1} C_{r+1} : { }^{n} C_{r} : { }^{n-1} C_{r-1} = 55 : 35 : 21$,then $2n + 5r$ is equal to:

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to $.......$.

The number of different words that can be formed from the letters of the word $SUCCESS$ in which the two $C$ are together but no two $S$ are together are: