Correct option: D.$\frac{{n(n\,\,1)\,(n\, + \,2)}}{6}$
d
$\sum\limits_{i\, = \,1}^n {\sum\limits_{j\, = \,1}^i {\sum\limits_{k = \,1}^j {1\,\, = } \,} } \,$
$\sum\limits_{i\, = \,1}^n {\sum\limits_{j\, = \,1}^i {j\,\,\, = \,\,\sum\limits_{i = \,1}^n {\frac{{i\;(i\, + \,1)}}{2}\,} \, = \,\,\frac{1}{2}\,\left[ {\sum\limits_{i\, = \,1}^n {{i^2}} \, + \,\sum\limits_{i\, = \,1}^n i } \right]} } $
$ = \,\frac{1}{2}\,\left[ {\frac{{n(n\, + \,1)\,(2n\, + \,1)}}{6}\,\, + \,\,\frac{{n\,(n\, + \,1)}}{2}} \right]$
$\, = \,\frac{{n\,(n\, + \,1)\,(n\, + \,2)}}{6}$