Prove that nCrn - 1Cr- 1 - nr where 1 - r - n

To prove: ^nC_r^(n 1)C_r 1= nr Solution: ^nC_r^(n 1)C_r 1=n!(n r)!r!(n 1)!(n 1 (r 1))!(r 1)! =n!(n r)!r!×(n r)!(r 1)!(n 1)! =n!r!×(r 1)!(n 1)! ...

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Standard XII

Mathematics

Combination

Prove that ...

Question

Prove that $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$ where $1 \leq r \leq n$

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Solution

To prove:
$\frac{^{n} C_{r}}{^{(n - 1)} C_{r - 1}} = \frac{n}{r}$
Solution:
$\frac{^{n} C_{r}}{^{(n - 1)} C_{r - 1}} = \frac{\frac{n!}{(n - r)! r!}}{\frac{(n - 1)!}{(n - 1 - (r - 1))! (r - 1)!}}$
$= \frac{n!}{(n - r)! r!} \times \frac{(n - r)! (r - 1)!}{(n - 1)!}$
$= \frac{n!}{r!} \times \frac{(r - 1)!}{(n - 1)!}$
$= \frac{n (n - 1)!}{r (r - 1)!} \times \frac{(r - 1)!}{(n - 1)!}$
$= \frac{n}{r}$
Hence, $\frac{^{n} C_{r}}{^{(n - 1)} C_{r - 1}} = \frac{n}{r}$

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Q. Prove that

\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}

when

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^{n} C_{r - 1} +^{n} C_{r} =^{n + 1} C_{r}

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Prove that nCrn−1Cr−1=nr where 1≤r≤n

To prove:nCr(n−1)Cr−1=nrSolution:nCr(n−1)Cr−1=n!(n−r)!r!(n−1)!(n−1−(r−1))!(r−1)!=n!(n−r)!r!×(n−r)!(r−1)!(n−1)!=n!r!×(r−1)!(n−1)!=n(n−1)!r(r−1)!×(r−1)!(n−1)!=nrHence, nCr(n−1)Cr−1=nr

Prove that $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$ where $1 \leq r \leq n$