If $^{43}C_{r-6} = ^{43}C_{3r+1}$,then the value of $r$ is

The number of ways in which $16$ identical oranges can be distributed among $4$ children such that each child gets at least one orange is:

If $^8C_r = ^8C_{r+2}$,then the value of $^rC_2$ is

The value of ${}^{50}C_4 + \sum_{r=1}^{6} {}^{56-r}C_3$ is

There are two urns. Urn $A$ contains $3$ distinct red balls,and Urn $B$ contains $9$ blue balls. If two balls are selected at random from each urn and swapped,in how many ways can this operation be performed?

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