In a football championship,there were played $153$ matches. Every team played one match with each other. The number of teams participating in the championship is

The solution set of ${ }^5 C_{x-1} > 2 \cdot { }^5 C_x$ is

In a lottery,a person chooses six different natural numbers at random from $1$ to $20$. If these six numbers match the six numbers already fixed by the lottery committee,he wins the prize. What is the probability of winning the prize in the game? [Hint: The order of the numbers is not important.]

$A$ team of $11$ players is to be selected from $22$ players. If $2$ specific players must be included in every team and $4$ specific players must always be excluded,in how many ways can this selection be made?

$A$ man has $7$ friends. In how many ways can he invite one or more of them for a tea party?

If $20$ identical books are to be distributed among $4$ persons such that each person receives at least one book,in how many ways can this be done?