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

Question

Philoid · Accepted Answer

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)!.