The number of non-negative integer solutions of the equations $6x + 4y + z = 200$ and $x + y + z = 100$ is

Let $x$ denote the number of ways of selecting at least one ball from a bag containing $3$ identical red balls,$4$ identical blue balls,and $5$ identical green balls. Let $y$ denote the number of ways in which a student can fail in an examination,when they have to write the examination in $5$ different subjects. Then $x+y=$

If the number of all possible permutations of the letters of the word $MATHEMATICS$ in which the repeated letters are not together is $90(X)$,then $X=$

The number of different words that can be formed from the letters of the word "$INTERMEDIATE$" such that two vowels never come together,is

The digit in the unit place of the number $(183!) + (3^{183})$ is

Find the remainder when $(1!)^2 + (2!)^2 + (3!)^2 + \dots + (100!)^2$ is divided by $10^2$.