If $$\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\,\left( {{}^{60}{C_{20}}} \right) = {m \over n}{}^{60}{C_{20}}$$ m and n are coprime, then m + n is equal to ___________.

<p>Here property used is</p> <p>${}^n{C_r} + {}^n{C_{r + 1}} = {}^{n + 1}{C_{r + 1}}$</p> <p>Given, $${}^{40}{C_0} + {}^{41}{C_1} + {}^{42}{C_2} + \,\,....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}{}^{60}{C_{20}}$$</p> <p>As ${}^{40}{C_0} = {}^{41}{C_0} = 1$</p> <p>So, we replace ${}^{40}{C_0}$ with ${}^{41}{C_0}$.</p> <p>$$ \Rightarrow {}^{41}{C_0} + {}^{41}{C_1} + {}^{42}{C_2} + \,\,.....\,\,\, + \,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$$</p> <p>$$ \Rightarrow {}^{42}{C_1} + {}^{42}{C_2} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$$</p> <p>$$ \Rightarrow {}^{43}{C_2} + {}^{43}{C_3} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$$</p> <p>$$ \Rightarrow {}^{44}{C_3} + {}^{44}{C_4} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$$</p> <p>$$ \Rightarrow {}^{45}{C_4} + {}^{45}{C_5} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$$</p> <p>$\vdots$</p> <p>$\Rightarrow {}^{60}{C_{19}} + {}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$</p> <p>$\Rightarrow {}^{61}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}$</p> <p>$\Rightarrow {{61!} \over {20!\,41!}} = {m \over n}\,.\,{{60!} \over {20!\,40!}}$</p> <p>$\Rightarrow {{61} \over {41}} = {m \over n}$</p> <p>$\therefore$ m = 61 and n = 41</p> <p>$\therefore$ m + n = 61 + 41 = 102</p>