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Applications of Colloids

here r / nCr ...

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here r/nCr=(n-r)/nCn-r how it is possble because its true that nCr=nCn-r but how n-r=r as in numerator

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

^{n} C_{n - r} + 3^{n} C_{n - r + 1} + 3.^{n} C_{n - r + 2} +^{n} C_{n - r + 3} =^{x} C_{r}

x = n + k

k

Q. Assertion (A) : The sum of the last ten coeffcients in the expansion of

(1 + x)^{19}

, when expanded in ascending powers of

x

2^{18}

n C_{r} =^{n} C_{n - r} (\begin{matrix} n r > - 2 \end{matrix}) \forall n \in N

Q. Prove that :

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

$^{n} C_{r} \div^{n} C_{r - 1}$ =

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