Sum of number of terms in the expansion of 

${(a_1+a_2)}^{50},{(a_1+a_2+a_3)}^{50},{(a_1+a_2+a_3+a_4)}^{50},\mathrm{......}{(a_1+a_2+\mathrm{....}+a_{50})}^{50}$
 is equal to
 

$Sum{=}^{51}{C}_1{+}^{52}{C}_2{+}^{53}{C}_3+\mathrm{....}{+}^{99}{C}_{49}$

$=\text{coefficient\hspace{0.17em}of\hspace{0.17em}}x\text{\hspace{0.17em}in\hspace{0.17em}}[{(1+x)}^{51}+x^{-1}{(1+x)}^{52}+x^{-2}{(1+x)}^{53}+\mathrm{......}+x^{-48}{(1+x)}^{99}]$$=\text{coefficient\hspace{0.17em}of\hspace{0.17em}}x\text{\hspace{0.17em}in\hspace{0.17em}}\frac{{(1+x)}^{51}[{\left(\frac{1+x}{x}\right)}^{49}-1]}{(\frac{1+x}{x}-1)}$

$=\text{coefficient\hspace{0.17em}of\hspace{0.17em}}x\text{\hspace{0.17em}in}[x^{-48}{(1+x)}^{100}-x{(1+x)}^{51}]$

$={}^{100}{C}_{49}-1{=}^{100}{C}_{51}-1$