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Question

If x is a positive integer, then
∣∣ ∣ ∣∣x!(x+1)!(x+2)!(x+1)!(x+2)!(x+3)!(x+2)!(x+3)!(x+4)!∣∣ ∣ ∣∣ is equal to

A
2x!(x+1)!
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B
2x!(x+1)!(x+2)!
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C
2x!(x+3)!
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D
2(x+1)!(x+2)!(x+3)!
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Solution

The correct option is A 2x!(x+1)!(x+2)!
∣ ∣ ∣x!(x+1)!(x+2)!(x+1)!(x+2)!(x+3)!(x+2)!(x+3)!(x+4)!∣ ∣ ∣

= x!(x+1)!(x+2)!∣ ∣111x+111(x+2)(x+1)(x+2)×2(x+3)×2∣ ∣

=x!(x+1)!(x+2)![(2(x+3)2(x+2)]=2(x!)(x+1)!(x+2)!

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