Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 56042 by Tawa1 last updated on 09/Mar/19

Answered by math1967 last updated on 09/Mar/19

c_r ^n =((n!)/(r!(n−r)!))×(((n−r+1))/((n−r+1)))=((n!×(n−r+1))/(r×(r−1)!(n−r+1)!))   =((n!)/((r−1)!(n−r+1)!))×[((n−r+1)/r)]  = ^n c_(r−1) [(((n−r+1))/r)]

$$\overset{{n}} {{c}}_{{r}} =\frac{{n}!}{{r}!\left({n}−{r}\right)!}×\frac{\left({n}−{r}+\mathrm{1}\right)}{\left({n}−{r}+\mathrm{1}\right)}=\frac{{n}!×\left({n}−{r}+\mathrm{1}\right)}{{r}×\left({r}−\mathrm{1}\right)!\left({n}−{r}+\mathrm{1}\right)!}\: \\ $$$$=\frac{{n}!}{\left({r}−\mathrm{1}\right)!\left({n}−{r}+\mathrm{1}\right)!}×\left[\frac{{n}−{r}+\mathrm{1}}{{r}}\right] \\ $$$$=\overset{{n}} {\:}{c}_{{r}−\mathrm{1}} \left[\frac{\left({n}−{r}+\mathrm{1}\right)}{{r}}\right] \\ $$

Commented by Tawa1 last updated on 09/Mar/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Answered by math1967 last updated on 09/Mar/19

c_(r ) ^n + ^n c_(r−1) =((n!)/(r!(n−r)!)) +((n!)/((r−1)!(n−r+1)!))  =n!{(1/(r(r−1)!(n−r)!)) +(1/((r−1)!(n−r+1)(n−r)!))}  =((n!)/((r−1)!(n−r)!))×{((n−r+1+r)/(r(n−r+1)))}  =(((n+1)×n!)/(r(r−1)!(n−r+1)(n−r)!))=(((n+1)!)/(r!(n+1−r)!))  = ^(n+1) c_r

$$\overset{{n}} {{c}}_{{r}\:} +\overset{{n}} {\:}{c}_{{r}−\mathrm{1}} =\frac{{n}!}{{r}!\left({n}−{r}\right)!}\:+\frac{{n}!}{\left({r}−\mathrm{1}\right)!\left({n}−{r}+\mathrm{1}\right)!} \\ $$$$={n}!\left\{\frac{\mathrm{1}}{{r}\left({r}−\mathrm{1}\right)!\left({n}−{r}\right)!}\:+\frac{\mathrm{1}}{\left({r}−\mathrm{1}\right)!\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!}\right\} \\ $$$$=\frac{{n}!}{\left({r}−\mathrm{1}\right)!\left({n}−{r}\right)!}×\left\{\frac{{n}−{r}+\mathrm{1}+{r}}{{r}\left({n}−{r}+\mathrm{1}\right)}\right\} \\ $$$$=\frac{\left({n}+\mathrm{1}\right)×{n}!}{{r}\left({r}−\mathrm{1}\right)!\left({n}−{r}+\mathrm{1}\right)\left({n}−{r}\right)!}=\frac{\left({n}+\mathrm{1}\right)!}{{r}!\left({n}+\mathrm{1}−{r}\right)!} \\ $$$$=\overset{{n}+\mathrm{1}} {\:}{c}_{{r}} \\ $$

Commented by Tawa1 last updated on 09/Mar/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by math1967 last updated on 09/Mar/19

GOD IS  A CIRCLE WHOSE CENTRE  IS EVERYWHERE,BUT  WHOSE  CIRCUMFERENCE IS NO WHERE

$${GOD}\:{IS}\:\:{A}\:{CIRCLE}\:{WHOSE}\:{CENTRE} \\ $$$${IS}\:{EVERYWHERE},{BUT}\:\:{WHOSE} \\ $$$${CIRCUMFERENCE}\:{IS}\:{NO}\:{WHERE} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com