Q. 164.0( 4 Votes )
Prove that the product of 2n consecutive negative integers is divisible by (2n)!
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)!.
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