coefficient x^6 from expressi (2x+1)^(6 ) × (x^2 +x+(1/4))^4 ?


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Question Number 82658 by jagoll last updated on 23/Feb/20

$c o e f f i c i e n t x^{6} f r o m e x p r e s s i$ ${(2 x + 1)}^{6} \times {(x^{2} + x + \frac{1}{4})}^{4} ?$
Commented by mr W last updated on 23/Feb/20

$= 2^{6} {(x + \frac{1}{2})}^{6} {(x + \frac{1}{2})}^{8}$ $= 2^{6} {(x + \frac{1}{2})}^{14}$ $= 2^{6} Σ \frac{1}{2^{14 - k}} C_{k}^{14} x^{k}$ $k = 6 :$ $2^{6} \times \frac{1}{2^{14 - 6}} \times C_{6}^{14} = \frac{C_{6}^{14}}{2^{2}} = \frac{3003}{4}$
Commented by jagoll last updated on 23/Feb/20

$t h a n k y o u m i s t e r$
	Answered by TANMAY PANACEA last updated on 23/Feb/20
	$(2x+1)^6 ×(((4x^2 +4x+1)/4))^4 (((2x+1)^6 )/4^4 )×{(2x+1)^2 }^4 (((2x+1)^(14) )/4^4 ) now let r+1 th term contains x^6 ((14_C_r )/4^4 )×(2x)^(14−r) ×1^r so x^(14−r) =x^6 →r=8 14c_8 ×(1/4^4 )×2^6 ×x^6$
	${(2 x + 1)}^{6} \times {(\frac{4 x^{2} + 4 x + 1}{4})}^{4}$ $\frac{{(2 x + 1)}^{6}}{4^{4}} \times {{(2 x + 1)}^{2}}^{4}$ $\frac{{(2 x + 1)}^{14}}{4^{4}}$ $n o w l e t r + 1 t h t e r m c o n t a i n s x^{6}$ $\frac{14_{C_{r}}}{4^{4}} \times {(2 x)}^{14 - r} \times 1^{r}$ $s o x^{14 - r} = x^{6} \to r = 8$ $14 c_{8} \times \frac{1}{4^{4}} \times 2^{6} \times x^{6}$
	Commented by jagoll last updated on 23/Feb/20

	$\frac{14 \times 13 \times 12 \times 11 \times 9 \times 8}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \times 64$
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