The number of four-digit numbers formed by using the digits $0, 2, 4, 5$ (without repetition) which are not divisible by $5$ is:

The minimum value of $f(x) = |x - 1| + |2x - 1| + |3x - 1| + \dots + |119x - 1|$ occurs at $x$. Then $x$ is equal to

Assuming that the balls of the same color are identical,find the number of ways to select one or more balls from $10$ white,$9$ green,and $7$ black balls.

How many times does the digit $5$ appear when writing natural numbers from $1$ to $100$?

All possible $5$-digit numbers,each having $5$ distinct digits,are formed using the digits $\{1, 2, 3, 5, 6, 8\}$. Among them,the number of such numbers which are divisible by $3$ but not by $6$ is:

Let $a_1, a_2, \ldots, a_n$ be $n$ non-zero real numbers,of which $p$ are positive and the remaining are negative. The number of ordered pairs $(j, k)$ with $j < k$ for which $a_j a_k$ is positive is $55$. Similarly,the number of ordered pairs $(j, k)$ with $j < k$ for which $a_j a_k$ is negative is $50$. Then,the value of $p^2 + (n-p)^2$ is