The number of words that do not start and end with vowels,formed using all the letters of the word $'UNIVERSITY'$ such that all vowels are in alphabetical order,is

If four letters are chosen from the letters of the word $ASSIGNMENT$ and are arranged in all possible ways to form $4$-letter words (with or without meaning),then the total number of such words that can be formed 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 value of the expression ${ }^{47} C_4 + \sum_{j=1}^5 { }^{52-j} C_3$ is

The total number of two-digit numbers $n$ such that $3^{n} + 7^{n}$ is a multiple of $10$ is ..... .

The number $(49^{2}-4)(49^{3}-49)$ is divisible by (in $!$)