Question Number 66995 by Sayantan chakraborty last updated on 21/Aug/19
Commented by Sayantan chakraborty last updated on 21/Aug/19
$$\mathrm{Can}\:\mathrm{anybody}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{this}??? \\ $$
Commented by Prithwish sen last updated on 21/Aug/19
$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{p}} {\sum}}\left(\:^{\mathrm{n}} \mathrm{C}_{\mathrm{k}} \right)\left(^{\mathrm{m}} \mathrm{C}_{\mathrm{p}−\mathrm{k}} \right)=^{\mathrm{n}+\mathrm{m}} \mathrm{C}_{\mathrm{p}} \:\:\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{formula} \\ $$$$ \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{use}\:\mathrm{here}.\:\mathrm{The}\:\mathrm{formula}\:\mathrm{can}\:\mathrm{be}\:\mathrm{proved} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{method}\:\mathrm{of}\:\mathrm{induction}.\:\mathrm{Or}\:\mathrm{you}\:\mathrm{can}\:\mathrm{try} \\ $$$$\mathrm{it}\:\mathrm{by}\:\mathrm{taking}\:\mathrm{say}\:\boldsymbol{\mathrm{p}}=\mathrm{6},\:\boldsymbol{\mathrm{n}}=\mathrm{6},\:\boldsymbol{\mathrm{m}}=\mathrm{7} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{way}\:\mathrm{you}\:\mathrm{can}\:\mathrm{get}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{turns} \\ $$$$\mathrm{to}\: \\ $$$$\underset{\mathrm{i}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}\:\frac{\mathrm{k}−\mathrm{i}}{\mathrm{M}−\mathrm{k}}\: \\ $$$$\boldsymbol{\mathrm{Y}}\mathrm{ou}\:\mathrm{can}\:\mathrm{also}\:\mathrm{prove}\:\mathrm{it}\:\mathrm{by}\:\mathrm{using}\:\mathrm{Pascals}\:\mathrm{triangle}. \\ $$
Commented by Sayantan chakraborty last updated on 22/Aug/19
$$\mathrm{Thanks}\:\mathrm{a}\:\mathrm{lot}. \\ $$
Commented by Prithwish sen last updated on 22/Aug/19
$$\mathrm{welcome}. \\ $$