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

$A$ group of $9$ students,$s_1, s_2, \ldots, s_9$,is to be divided into three teams $X, Y$,and $Z$ of sizes $2, 3$,and $4$,respectively. Suppose that $s_1$ cannot be selected for team $X$,and $s_2$ cannot be selected for team $Y$. The number of ways to form such teams is:

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

If $\frac{1}{{ }^{5}C_{r}} + \frac{1}{{ }^{6}C_{r}} = \frac{1}{{ }^{4}C_{r}},$ then the value of $r$ is

The number of four-digit numbers that can be formed using all the digits from $1$ to $9$ (excluding zero) such that every number has exactly $2$ distinct digits is:

If $^{15}C_{3r} = ^{15}C_{r+3}$,find the value of $r$.