Let r and n be positive integers such that 1≤r≤n. Then prove the f

1.	Let r and n be positive integers such that 1≤r≤n. Then prove the following : (i) nCrnCr−1=n−r+1r (ii) nn−1Cr−1=(n−r+1)nCr−1 (iii) nCrn−1Cr−1=nr (iv) nCr+2nCr−1+nCr−2=n+2Cr
Answer» Let r and n be positive integers such that $1 \leq r \leq n .$ Then prove the following : (i) $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$ (ii) $n^{n - 1} C_{r - 1} = (n - r + 1)^{n} C_{r - 1}$ (iii) $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$ (iv) $^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}$

Let r and n be positive integers such that 1≤r≤n. Then prove the following : (i) nCrnCr−1=n−r+1r (ii) nn−1Cr−1=(n−r+1)nCr−1 (iii) nCrn−1Cr−1=nr (iv) nCr+2nCr−1+nCr−2=n+2Cr

Answer»

Let r and n be positive integers such that $1 \leq r \leq n .$ Then prove the following :

(i) $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$
(ii) $n^{n - 1} C_{r - 1} = (n - r + 1)^{n} C_{r - 1}$
(iii) $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$
(iv) $^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}$

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Let r and n be positive integers such that 1≤r≤n. Then prove the following : (i) nCrnCr−1=n−r+1r (ii) nn−1Cr−1=(n−r+1)nCr−1 (iii) nCrn−1Cr−1=nr (iv) nCr+2nCr−1+nCr−2=n+2Cr

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