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

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

thnks! 

Grade:11

1 Answers

Aman Bansal
592 Points
11 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:

    (a + b)^n = sum[k=0,n][(n over k)a^(n-k)b^k]   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

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:

    (n over k) = nCk = n!/[(n-k)!k!]

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:

     

    10_C_7 = 10! / [(10 - 7)! 7!] = 10! / [3!7!] = 120

     

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    Aman Bansal

    Askiitian Expert

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