How many numbers lying between $1000$ and $10000$ can be formed by using the digits $1, 3, 5, 6, 7, 8, 9$ if no digits are repeated?

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves exceeded the number of games that the men played with the women by $66$. The total number of participants is:

Find the number of all even $2$-digit numbers that do not have $0$ at the unit's place.

In how many ways can $6$ persons stand in a queue at a time?

$^nC_r + ^{n-1}C_r + ...... + ^rC_r$ =

Out of $10$ white,$9$ black and $7$ red balls,the number of ways in which selection of one or more balls can be made,is