In how many ways can $4$ balls be picked from $6$ black and $4$ green colored balls such that at least one black ball is selected?

In a shop,there are $5$ types of ice-creams available. $A$ child buys $6$ ice-creams.
$Statement-1$: The number of different ways the child can buy the $6$ ice-creams is $^{10}C_5$.
$Statement-2$: The number of different ways the child can buy the $6$ ice-creams is equal to the number of different ways of arranging $6$ $A$'s and $4$ $B$'s in a row.

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.

From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is

The number of solutions of the equation $x+y+z=10$, where $x, y$, and $z$ are positive integers, is:

$A$ country has $10$ smart cities. The government decides to connect all these cities by road. How many roads does the government need to construct so that every city is connected to every other city?