The number of all possible ways of distributing $8$ identical apples among $3$ persons is

$^nC_r + 2^nC_{r-1} + ^nC_{r-2} = $

If $n$ objects are arranged in a row,then the number of ways of selecting three of these objects such that no two of them are next to each other 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

The number of ways in which $3$ identical balls can be distributed into $7$ distinct bins is

The number of $4$-letter words,with or without meaning,each consisting of $2$ vowels and $2$ consonants,which can be formed from the letters of the word $UNIVERSE$ without repetition is $.........$.