The value of the expression ${ }^{47} C_4 + \sum_{j=1}^5 { }^{52-j} C_3$ is

$n$ objects are distributed at random among $n$ persons. The number of ways in which this can be done so that at least one of them will not get any object is

Four-digit numbers with all digits distinct are formed using the digits $1, 2, 3, 4, 5, 6, 7$ in all possible ways. If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than $3400$ among them,then $p: q=$

The number of ways of distributing $15$ apples to three persons $A, B, C$ such that $A$ and $C$ each get at least $2$ apples and $B$ gets at most $5$ apples is

$A$ question paper has $3$ parts and each part contains $4$ questions. The number of different ways in which a candidate can answer $8$ questions choosing at least two from each part is

Numbers between $1$ and $10,000$ are formed using the digits $2$ and $3$ exactly once and the digit $4$ twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of $4324$ and $324$ respectively,then $x-y=$