The sum of the series 8!+9!2!+10!3!+11!4!+…… up to 20 terms

7!+8!+9!2!+10!3!+11!4!+……−7!7![1+8!1!7!+9!7!2!+….+27!7!20!]−7!=7!(7C0+8C1+9C2+……..27C20)−7!=7!(28C20)−7!

The sum of the series 8!+9!2!+10!3!+11!4!+…… up to 20 terms

7!+8!+9!2!+10!3!+11!4!+……−7!7![1+8!1!7!+9!7!2!+….+27!7!20!]−7!=7!(7C0+8C1+9C2+……..27C20)−7!=7!(28C20)−7!

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The sum of the series $8! + \frac{9!}{2!} + \frac{10!}{3!} + \frac{11!}{4!} + \dots \dots$ up to 20 terms

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$7! (^{27} C_{19} - 1)$

$8! (^{28} C_{20} - 1)$

$8! (^{27} C_{20} - 1)$

$7! (^{28} C_{20} - 1)$

answer is C.

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Detailed Solution

$7! + 8! + \frac{9!}{2!} + \frac{10!}{3!} + \frac{11!}{4!} + \dots \dots - 7!$

$7! [1 + \frac{8!}{1! 7!} + \frac{9!}{7! 2!} + \dots . + \frac{27!}{7! 20!}] - 7!$

$= 7! (^{7} C_{0} +^{8} C_{1} +^{9} C_{2} + \dots \dots . .^{27} C_{20}) - 7! = 7! (^{28} C_{20}) - 7!$

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The sum of the series $8! + \frac{9!}{2!} + \frac{10!}{3!} + \frac{11!}{4!} + \dots \dots$ up to 20 terms