$^nC_r + ^nC_{r-1}$ is equal to

$A$ 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

$A$ person wants to climb a $n$-step staircase using one step or two steps. Let $C_n$ denote the number of ways of climbing the $n$-step staircase. Then $C_{18} + C_{19}$ equals

In a city,no two persons have an identical set of teeth,and there is no person without a tooth. Also,no person has more than $32$ teeth. If we disregard the shape and size of the teeth and consider only the presence or absence of a tooth at each of the $32$ positions,then the maximum population of the city is:

$^{14}C_4 + \sum_{j=1}^4 {^{18-j}C_3}$ is equal to

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