If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

The value of $^{15}C_{3} + ^{15}C_{13}$ is

The least value of the natural number $n$ satisfying $C(n, 5)+C(n, 6)>C(n+1, 5)$ is

The number of groups that can be made from $5$ different green balls,$4$ different blue balls,and $3$ different red balls,if at least $1$ green and $1$ blue ball must be included.

The number of integers between $10$ and $10,000$ such that in every integer every digit is greater than its immediate preceding digit,is

The number of ordered pairs $(r, k)$ for which $6 \cdot ^{35}C_{r} = (k^{2} - 3) \cdot ^{36}C_{r+1}$,where $k$ is an integer,is