This kind of loops can be thought similar to ball and buckets problem where range of i (1 to n) corresponds to n number of buckets and r number of loop variables corresponds to r balls.
Total no of loop executions are$\large _{}^{n+r-1}\textrm{C}_{r}$ .(Check from Rosen page no 427 for more explanation)
Now in this question, total number of loop executions = $\large _{}^{n+3-1}\textrm{C}_{3}$
but we have counted extra executions cause k = j+1 to n not j to n , So we have to subtract those cases which is $\large _{}^{n+2-1}\textrm{C}_{2}$ (cause we have to subtract those cases where i <= j = k so j,k can be thought as one variable)
So total number of multiplications to be performed = $\large _{}^{n+2}\textrm{C}_{3}$ – $\large _{}^{n+1}\textrm{C}_{2}$ = $\large \frac{(n+1).n.(n-1)}{6}$