The number of all four-digit numbers which do not have four distinct digits is

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$,if all digits are to appear in the same number at least once,is

The value of $\sum\limits_{r = 1}^{15} {{r^2} \left( \frac{^{15}C_r}{^{15}C_{r - 1}} \right)}$ is equal to

Let $x, y, z$ be three non-negative integers such that $x+y+z=10$. The maximum possible value of $xyz+xy+yz+zx$ is

The number of different $5$-digit numbers greater than $50000$ that can be formed using the digits $0, 1, 2, 3, 4, 5, 6, 7$,such that the sum of their first and last digits is not more than $8$,is:

If a number is chosen at random from the four-digit numbers formed by using the digits $0, 1, 2, 3, 4, 6$ without repetition,then the probability that it is divisible by $4$ is