If ( ^40C_0 ) + ( ^41C_1 ) + ( ^42C_2 ) + ..... + ( ^60C_20 ) = m / n^60C_20 m and n are coprime, then m + n is equal to ___________.

Here property used is ^nC_r + ^nC_r + 1 = ^n + 1C_r + 1 Given, ^40C_0 + ^41C_1 + ^42C_2 + .... + ^60C_20 = m / n^60C_20 As ^40C_0 = ^41C_0 = 1 So, we replace ^40C_0 with ^41C_0 => ^41C_0 + ^41C_1 + ^42C_2 + ..... + ^60C_20 = m / n . ^60C_20 => ^42C_1 + ^42C_2 + ..... + ^60C_20 = m / n . ^60C_20 => ^43C_2 + ^43C_3 + ..... + ^60C_20 = m / n . ^60C_20 => ^44C_3 + ^44C_4 + ..... + ^60C_20 = m / n . ^60C_20 => ^45C_4 + ^45C_5 + ..... + ^60C_20 = m / n . ^60C_20 => ^60C_19 + ^60C_20 = m / n . ^60C_20 => ^61C_20 = m / n . ^60C_20 => 61! / 20! 41! = m / n . 60! / 20! 40! => 61 / 41 = m / n Therefore m = 61 and n = 41 Therefore m + n = 61 + 41 = 102

If 40C0 41C1 42C2 60C20 m n60C20 m and n are coprime th... | null (Mathematics)

Here property used is

{}^n{C_r} + {}^n{C_{r + 1}} = {}^{n + 1}{C_{r + 1}}

Given,

{}^{40}{C_0} + {}^{41}{C_1} + {}^{42}{C_2} + \,\,....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}{}^{60}{C_{20}}

\text{As} \ \ {}^{40}{C_0} = {}^{41}{C_0} = 1

\text{So, we replace} \ {}^{40}{C_0} \ \text{with} \ {}^{41}{C_0}

\Rightarrow {}^{41}{C_0} + {}^{41}{C_1} + {}^{42}{C_2} + \,\,.....\,\,\, + \,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\Rightarrow {}^{42}{C_1} + {}^{42}{C_2} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\Rightarrow {}^{43}{C_2} + {}^{43}{C_3} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\Rightarrow {}^{44}{C_3} + {}^{44}{C_4} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\Rightarrow {}^{45}{C_4} + {}^{45}{C_5} + \,\,.....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\vdots

\Rightarrow {}^{60}{C_{19}} + {}^{60}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\Rightarrow {}^{61}{C_{20}} = {m \over n}\,.\,{}^{60}{C_{20}}

\Rightarrow {{61!} \over {20!\,41!}} = {m \over n}\,.\,{{60!} \over {20!\,40!}}

\Rightarrow {{61} \over {41}} = {m \over n}

$\therefore$ m = 61 and n = 41

$\therefore$ m + n = 61 + 41 = 102

Explanation

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