In the product $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{100})$,when written in ascending powers of $x$,the highest exponent of $x$ is . . . . . . .

$5$ boys and $6$ girls are arranged in all possible ways. Let $X$ denote the number of linear arrangements in which no two boys sit together and $Y$ denote the number of linear arrangements in which no two girls sit together. If $Z$ denotes the number of ways of arranging all of them around a circular table such that no two boys sit together,then $X: Y: Z=$

Letters in the word $HULULULU$ are rearranged. The probability of all three $L$ being together 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:

If ${ }^{1} P_{1}+2 \cdot{ }^{2} P_{2}+3 \cdot{ }^{3} P_{3}+\ldots+15 \cdot{ }^{15} P_{15}={ }^{q} P_{r}-s$,where $0 \leq s \leq 1$,then ${ }^{q+s} C_{r-s}$ is equal to .... .

For all $n \in N$,the product $(n+24)(n+25)(n+26)(n+27)$ is always divisible by: