1)Value of 20!+((21!)/(1!))+((22!)/(2!))+....+((60!)/(40!)) is ? 2) Sum of all solutions of eq^n : cos 3θ=sin 2θ in interval [−(π/2),(π/2)] is ?


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Question Number 58406 by rahul 19 last updated on 22/Apr/19

$1) V a l u e o f 20! + \frac{21!}{1!} + \frac{22!}{2!} + . . . . + \frac{60!}{40!} i s ?$ $2) S u m o f a l l s o l u t i o n s o f {e q}^{n} :$ $\cos 3 θ = \sin 2 θ i n i n t e r v a l [- \frac{π}{2}, \frac{π}{2}] i s ?$
	Answered by MJS last updated on 23/Apr/19

	$(1)$ $29 623 769 613 544 887 392 514 571 468 800 000$
	Commented by rahul 19 last updated on 23/Apr/19

	$S i r, k i n d l y s h o w y o u r w o r k i n g . . .$ $A n s \to 20!^{61} C_{21} .$
	Commented by MJS last updated on 23/Apr/19

	$to be honest, I just computed it . . .$
	Commented by rahul 19 last updated on 23/Apr/19

	$h o w m u c h t i m e d i d i t t a k e ?$
	Answered by MJS last updated on 23/Apr/19

	$(2)$ $\cos 3 θ = \sin 2 θ$ $\cos θ (1 - 2 \sin θ) (1 + 2 \sin θ) = 2 \cos θ \sin θ$ $\cos θ = 0 \Rightarrow θ = - \frac{π}{2} \lor θ = \frac{π}{2}$ ${4 \sin}^{2} θ - 2 \sin θ - 1 = 0$ $\sin θ = - \frac{1}{4} \pm \frac{\sqrt{5}}{4} \Rightarrow θ = - \frac{3 π}{10} \lor θ = \frac{π}{10}$ $\Rightarrow answer is - \frac{π}{5}$
	Commented by rahul 19 last updated on 23/Apr/19

	$t h a n k s s i r!$
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