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Question: \(\frac{- 462a^{5}}{b^{6}}\)=...

−462a5b6\frac{- 462a^{5}}{b^{6}}=

A

−462a6b5\frac{- 462a^{6}}{b^{5}}

B

∑m=0100100Cm(x−3)100−m.2m\sum_{m = 0}^{100}{100C_{m}(x - 3)^{100 - m}}.2^{m}

C

x53x^{53}

D

None of these

Answer

x53x^{53}

Explanation

Solution

Putting the values of nCr(3r−2n)nC_{r}(3^{r} - 2^{n}) we get

nCr(3r+2n−r)nC_{r}(3^{r} + 2^{n - r})

=(α2x2−2α x+1)51(\alpha^{2}x^{2} - 2\alpha\ x + 1)^{51}

Put α\alpha = N

= x+y=1x + y = 1

∑r=0nr2nCrxryn−r\sum_{r = 0}^{n}{{r^{2}}^{n}C_{r}x^{r}y^{n - r}}

nx(x+yn)nx(x + yn) nx(nx+y)nx(nx + y)

Trick : Put n=1, then 4nC0+4nC4+4nC8+....+4nC4n4nC_{0} +^{4n}C_{4} +^{4n}C_{8} + .... +^{4n}C_{4n}

At n=2, 24n−2+(−1)n22n−12^{4n - 2} + ( - 1)^{n}2^{2n - 1}

Also (3) 24n−2+22n−12^{4n - 2} + 2^{2n - 1}