Group $A$ consists of $7$ boys and $3$ girls,while group $B$ consists of $6$ boys and $5$ girls. The number of ways,$4$ boys and $4$ girls can be invited for a picnic if $5$ of them must be from group $A$ and the remaining $3$ from group $B$,is equal to:

Let $n = 1! + 4! + 7! + \ldots + 400!$. Then the ten's digit of $n$ is

Words of length $10$ are formed by using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated and $y$ be the number of such words where exactly two letters are repeated twice and no other letter is repeated,then the value of $\frac{y}{x}$ is

$A$ pack contains $n$ cards numbered from $1$ to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on the removed cards is $k$,then $k - 20 =$

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:

Ten students are seated at random in a row. The probability that two particular students are not seated side by side is