How many $3$-digit numbers can be formed from the digits $2, 3, 5, 6, 7,$ and $9$ which are divisible by $5$,given that no digit is repeated?

The value of $^{15}C_3 + ^{15}C_{13}$ is

Let $A_1, A_2, ......, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, ......, 11$. One hundred gold coins were won by the team. These coins are to be distributed among the players such that each player $A_i$ gets at least $i$ coins,plus an additional coin (i.e.,at least $i+1$ coins). Additionally,the captain (assume $A_1$) and vice-captain (assume $A_2$) must receive at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. In how many different ways can these coins be distributed?

There are $5$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back is:

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is

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