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Railways in India and the Challenges Faced by It

find n and r ...

Question

find n and r if ncr:ncr+1:ncr+2=1:2:3.

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^{n} C_{r - 1} :^{n} C_{r} :^{n} C_{r + 1} = 2 : 3 : 4

(n, r)

^{n} C_{r} :^{n} C_{r + 1} :^{n} C_{r + 2} = 3 : 4 : 5

, then the value of

2 n + 3 r

\frac{^{n} C_{r}}{^{n} C_{r + 1}} = \frac{1}{2}

\frac{^{n} C_{r + 1}}{^{n} C_{r + 2}} = \frac{2}{3}

; determine the values of

n

r

\frac{^{n} C_{r} + 3^{n} C_{r + 1} + 3^{n} C_{r + 2} +^{n} C_{r + 3}}{^{n} C_{r} + 4^{n} C_{r + 1} + 6^{n} C_{r + 2} + 4^{n} C_{r + 3} +^{n} C_{r + 4}} = \frac{r + k}{n + k}

, then the value of

k

\frac{^{n} C_{r} + 4^{n} C_{r + 1} + 6^{n} C_{r + 2} + 4^{n} C_{r + 3} +^{n} C_{r + 4}}{^{n} C_{r} + 3^{n} C_{r + 1} + 3^{n} C_{r + 2} +^{n} C_{r + 3}} = \frac{n + k}{r + k}

, then the value of k is

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