$A$ committee of $3$ persons is to be constituted from a group of $2$ men and $3$ women. In how many ways can this be done? How many of these committees would consist of $1$ man and $2$ women?

In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?

In how many ways can a committee of $5$ members be formed from $6$ men and $4$ women such that the committee contains at least one woman?

If $^nC_{r-2} = 36$,$^nC_{r-1} = 84$,and $^nC_r = 126$,then the value of $^nC_{2r}$ is

Let $A_1, A_2, \dots, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, \dots, 11$. One hundred gold coins were won by the team in the final match. These coins are to be distributed among the players such that each player $A_i$ gets at least $i+1$ coins,except for the captain and vice-captain who get at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. If the captain is $A_1$ and the vice-captain is $A_2$,in how many different ways can these coins be distributed?

Six '$+$' and four '$-$' signs are to be placed in a straight line so that no two '$-$' signs come together. Then the total number of ways is: