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Let r and n be positive integers such th...

Let `r` and `n` be positive integers such that `1 leq r leq n`. Then prove the following: `n+n-1 C_{r-1}=(n-r+1){ }^{n} C_{r-1}`

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LHS `=n_{.}^{n-1} C_{r-1}`

`=frac{n(n-1) !}{(r-1) !(n-1-r+1) !}`

`=frac{n !}{(r-1) !(n-r) !}`

RHS = `(n-r+1)^{n} C_{r}`

`=(n-r+1) frac{n !}{(r-1) !(n-r+1) !}`

`=(n-r+1) frac{n !}{(r-1) !(n-r+1)(n-r) !}`

`=frac{n !}{(r-1) !(n-r) !}`

`therefore` LHS =RHS

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