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# Show that (nC0 )nC0 -((n+1)C 1) n 1 +((n+2)C2) nC2-..........= (-1)^nthnks!

Aman Bansal
592 Points
8 years ago

Dear Swapnil,

The Binomial Theorem is a quick way (okay, it''s a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody figured out a formula for this expansion, and we can plug the binomial 3x – 2 and the power 10 into that formula to get that expanded (multiplied-out) form.

The formal expression of the Binomial Theorem is as follows:

Yeah, I know; that formula never helped me much, either. And it doesn''t help that different texts use different notations to mean the same thing. The parenthetical bit above has these equivalents:

Recall that the factorial notation "n!" means " the product of all the whole numbers between 1 and n", so, for instance, 6! = 1×2×3×4×5×6. Then the notation "10C7" (often pronounced as "ten, choose seven") means:

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Thanks

Aman Bansal