Prove that the product of 2n consecutive negative integers is divisible by (2n)!

Let us assume the negative consecutive integers are –1, – 2,......, – (2n)

Let M be the product of the negative integers,

M = ( – 1).( – 2)( – 3)......( – 2n + 1).( – 2n)

M = ( – 1)2n(1.2.3.......(2n – 1).(2n))

M = 1.2.3......……(2n – 1).(2n)

We know that, n! = n(n – 1)(n – 2)…………2.1

M = (2n)!

M is divisible by (2n)!.

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