BinoMial - II

If - Z= \frac{1}{^nC_0} + \frac{1}{^nC_1} +....+ \frac{1}{^nC_n}

Then, find the value of -

Z= \frac{1}{^nC_1} + \frac{2}{^nC_2} + \frac{3}{^nC_3}+....+ \frac{n}{^nC_n}

in terms of "n" and "Z".

2 Answers

Z = 1 / ⁿC₀ + 1/ ⁿC₁ + ..........+ 1 / ⁿC_n

= 1ⁿ C _r

Now , Z ' = 1/ ⁿ C₁ + 2 /ⁿ C ₂ + 3 / ⁿC₃+ ............+ n / ⁿ C_n

= rⁿ c_r ................(i)
Now replace r by (n-r) in the eqn, we get

Z ' = (n-r)ⁿ C _n-r

= (n-r)ⁿ c _r .............(ii) [bcoz ⁿ C_r=ⁿ C _n-r ]

Now adding eqn (i) and (ii), we get
2 Z ' = n 1ⁿ c_r = n Z (bcoz Z = 1ⁿ c _r)

Therefore, Z ' = (1/2) n Z

Perfectly done Tushar...thnx [1]

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