The coefficient of $x^{50}$ in the expansion of $\sum _{k=0}^{100}{100}_{C_k}.{\left(x-2\right)}^{100-k}{3}^k$  is also equal to.

Number of dissimilar terms in   ${\left(x_1+x_2+x_3+\mathrm{.....}{x}_{50}\right)}^{51}$

$\frac{\mathrm{2.6.10.14.......198}}{50!}$

Number of ways in which 50 identical books can be distributed in 100 students, if each student can get at most one book.

Number of ways in which 100 different white balls and 50 identical red balls can be arranged in a circle, if no  two red balls are together.