Question Number 54646 by gunawan last updated on 08/Feb/19
$$\mathrm{Such}\:\mathrm{That} \\ $$$$\mathrm{a}.\:_{{n}+\mathrm{1}} {C}_{{r}} =\frac{\left({n}+\mathrm{1}\right).\:_{{n}} {C}_{{r}} }{\left({n}−{r}+\mathrm{1}\right)} \\ $$$$\mathrm{b}.\:_{{n}} {C}_{\mathrm{0}} +_{{n}} {C}_{\mathrm{2}} +_{{n}} {C}_{\mathrm{4}…} =_{{n}} {C}_{\mathrm{1}} +_{{n}} {C}_{\mathrm{3}} +_{{n}} {C}_{\mathrm{5}…} =\mathrm{2}^{{n}−\mathrm{1}} \\ $$$$ \\ $$
Answered by Kunal12588 last updated on 08/Feb/19
$${LHS}=^{{n}+\mathrm{1}} {C}_{{r}} =\frac{\left({n}+\mathrm{1}\right)!}{{r}!\left({n}+\mathrm{1}−{r}\right)!} \\ $$$$=\frac{\left({n}+\mathrm{1}\right){n}!}{{r}!\left({n}+\mathrm{1}−{r}\right)\left({n}−{r}\right)!}=\frac{{n}+\mathrm{1}}{{n}−{r}+\mathrm{1}}×\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$$$=\frac{\left({n}+\mathrm{1}\right)\:\:^{{n}} {C}_{{r}} }{\left({n}+\mathrm{1}−{r}\right)}={RHS}\:\:\:{proved} \\ $$
Commented by gunawan last updated on 08/Feb/19
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{Sir} \\ $$
Answered by Kunal12588 last updated on 10/Feb/19
$${b}.\:\:{from}\:{binimial}\:{expansion} \\ $$$$\mathrm{0}^{{n}} =\left(\mathrm{1}−\mathrm{1}\right)^{{n}} \\ $$$$\Rightarrow\mathrm{0}=^{{n}} {C}_{\mathrm{0}} −^{{n}} {C}_{\mathrm{1}} +^{{n}} {C}_{\mathrm{2}} −…..+\left(−\mathrm{1}\right)^{{n}} \:\centerdot\:^{{n}} {C}_{{n}} \\ $$$$\Rightarrow^{{n}} {C}_{\mathrm{0}} +^{{n}} {C}_{\mathrm{2}} +^{{n}} {C}_{\mathrm{4}} +….=^{{n}} {C}_{\mathrm{1}} +^{{n}} {C}_{\mathrm{3}} +^{{n}} {C}_{\mathrm{5}} +….\:\:\:\:{eq}\left(\mathrm{1}\right) \\ $$$${LHS}\:\:{of}\:{eq}\left(\mathrm{1}\right) \\ $$$$\:^{{n}} {C}_{\mathrm{0}} +^{{n}} {C}_{\mathrm{2}} +^{{n}} {C}_{\mathrm{4}} +…. \\ $$$$=^{{n}−\mathrm{1}} {C}_{\mathrm{0}} +^{{n}−\mathrm{1}} {C}_{\mathrm{1}} +^{{n}−\mathrm{1}} {C}_{\mathrm{2}} +^{{n}−\mathrm{1}} {C}_{\mathrm{3}} +^{{n}−\mathrm{1}} {C}_{\mathrm{4}} +… \\ $$$$=\mathrm{2}^{{n}−\mathrm{1}} \:\:\:\:{proved} \\ $$$${I}\:{used}\:{the}\:{identities}\:\left(\mathrm{1}\right)\:{and}\:\left(\mathrm{3}\right)\:\:{from}\: \\ $$$${question}\:\mathrm{54740}\:{see}\:{carefully} \\ $$