[COMBINATION]Prove that nCr/nCr+nCr+1 = r+1/n+1
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As posted, the equation is as follows:
. . . . .\(\displaystyle \dfrac{{}_nC_r}{{}_nC_r}\, +\, {}_nC_r\, +\, 1\, =\, r\, +\, \dfrac{1}{n}\, +\, 1\)
I'm fairly certain that this is not what you mean. For instance, the right-hand side is almost certainly meant to be as follows:
. . . . .\(\displaystyle \dfrac{r\, +\, 1}{n\, +\, 1}\)
...which one would type as "(r + 1)/(n + 1)". (The grouping symbols make all the difference!)
Please review this article on formatting math as text, and reply with corrections. When you reply, please include a clear listing of your thoughts and efforts so far, starting with what you set up as your formulas for the left-hand side. Thank you!
As posted, the equation is as follows:
. . . . .\(\displaystyle \dfrac{{}_nC_r}{{}_nC_r}\, +\, {}_nC_r\, +\, 1\, =\, r\, +\, \dfrac{1}{n}\, +\, 1\)
I'm fairly certain that this is not what you mean. For instance, the right-hand side is almost certainly meant to be as follows:
. . . . .\(\displaystyle \dfrac{r\, +\, 1}{n\, +\, 1}\)
...which one would type as "(r + 1)/(n + 1)". (The grouping symbols make all the difference!)
Please review this article on formatting math as text, and reply with corrections. When you reply, please include a clear listing of your thoughts and efforts so far, starting with what you set up as your formulas for the left-hand side. Thank you!
Sorry for writing the problem in too bad way that's because i couldn't write it as math text even the article didn't help mee i didn't understand it actually so i took a photo for the problem maybe it would help
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Sorry for writing the problem in too bad way that's because i couldn't write it as math text even the article didn't help mee i didn't understand it actually so i took a photo for the problem maybe it would help
Hint:
\(\displaystyle \displaystyle{^nC_r \ = \ \dfrac{n!}{r! \ * \ (n-r)!}}\)
continue .... (or wait for Soroban to give you a camera-ready solution)
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Anotherway:
\(\displaystyle \displaystyle{\dfrac{^nC_r \ + \ ^nC_{r+1}}{^nC_r}}\)
\(\displaystyle \displaystyle{ = \ 1 \ + \ \dfrac{^nC_{r+1}}{^nC_r}}\)
\(\displaystyle \displaystyle{ = \ 1 \ + \ \dfrac{r! \ * \ (n-r)!}{(r+1)! \ * [(n-r) - 1]!}}\)
\(\displaystyle \displaystyle{ = \ 1 \ + \ \dfrac{(n-r)}{(r+1)}}\)
\(\displaystyle \displaystyle{ = \dfrac{(n+1)}{(r+1)}}\)
so
\(\displaystyle \displaystyle{\dfrac{^nC_r}{^nC_r \ + \ ^nC_{r+1}} \ = \ \dfrac{(r+1)}{(n+1)}} \)