There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

Given $5$ different green toys,$4$ different blue toys,and $3$ different red toys,how many combinations of toys can be chosen taking at least one green and one blue toy?

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

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

$\binom{n}{r+1} + 2\binom{n}{r} + \binom{n}{r-1} = \dots$

If $\binom{15}{3r} = \binom{15}{r+3}$,then $r = \dots$