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Prove that C0Cr+C1 C(r+1)+ C2 C(r+2)+.....

Prove that `C_0C_r+C_1 C_(r+1)+ C_2 C_(r+2)+...............+c_(n-r) C_n=((2n)!)/((n-r)!(n+r)!)`

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If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

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If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : (1+C_1/C_0)(1+C_2/C_1)(1+C_3/C_2).... (1+C_n/ C_(n-1))=((n+1)^n)/(n!) .

MODERN PUBLICATION-BINOMIAL THEOREM-EXERCISE
  1. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : 2C0...

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  2. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : C0C...

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  3. Prove that C0Cr+C1 C(r+1)+ C2 C(r+2)+...............+c(n-r) Cn=((2n)!...

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  4. Find the coefficient of x^(n-r) in the expansion of (x+1)^n (1+x)^n. ...

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  5. Find the coefficient of x^4 in the expansion of (1+x)^n (1-x)^n. Deduc...

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  6. Write and simplify the term involving a^2b^5 in (a-2b)^4 (a+b)^3.

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  7. Expand using binomial theorem (1+x/2-2/x)^(4), x ne 0

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  8. Find an approximation of (0.99)^5 using the first three terms of its e...

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  9. If 6th, 7th, 8th and 9th terms of (x+y)^n are a, b,c and d respectivel...

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  10. Find the coefficient of x^10 in the expansion of : (1+x+x^2+x^3 +........

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  11. If kr is the coefficient of y^(r-1) in the expansion of (1+2y)^10, in ...

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  12. If x^p occurs in the expansion of (x^2+1/x)^(2n), prove that its coeff...

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  13. Determine the term independent of x in the expansion of (1+x+x^(-2)+x^...

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  14. Find the value of (a^2+ sqrt(a^2-1))^4+(a^2- sqrt(a^2-1))^4.

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  15. If fourth term in the expansion of : {sqrt(x^(1/(logx+1)))+x^(1/12)}^6...

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  16. Find the value of x for which the sixth term of : (sqrt(2^(log(10-3^x)...

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  17. If in the expansion of (1-x)^(2n-1), the coefficient of x^r is denoted...

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  18. If C1,C2,C3,C4 are the coefficients of any four consecutive terms in t...

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  19. For the expansion of (1 + x)^n= C0+C1x+C2x^2+......+ Cn x^n, show that...

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  20. 7! div 5! is :

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