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Give that : C1+2C2 x+3C3 x^2+.....+2n. ...

Give that : `C_1+2C_2 x+3C_3 x^2+.....+2n. C_(2n). x^(2n-1)=2n(1+x)^(2n-1)`, where `C_r=((2n)!)/(r !(2n-r)!)`, r=0,1,2, ,2n , then prove that `C_1^2-2C_2^2+3C_3^2-..........-2n C_(2n)^2=(-1)^n . nC_n` .

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MODERN PUBLICATION-BINOMIAL THEOREM-EXERCISE
  1. The letters of the word COCHIN are permuted and all the permutations a...

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  2. The sum of series .^(20)C0-^(20)C1+^(20)C2-^(20)C3+....+^(20)C 10 is

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  3. In the binomial expansion of (a -b)^n, nge5, the sum of the 5th and 6t...

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  4. How many different words can be formed by jumbling the letters of the ...

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  5. The number of 7-digit integers, with sum of the digits equal to 10 and...

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  6. The remainder left out when 8^(2n) - (62)^(2n +1) is divided by 9 , i...

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  7. From 6 different novels and 5 different dictionaries, 4 novels and 1 d...

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  8. There are two urns. Urn A has 4 distinct red balls and urn B has 5 dis...

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  9. There are 10 points in a plane, out of these 6 are collinear. If N is ...

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  10. For r = 0 ,1,2,…, 10 , letA(r), B(r) "and " C(r) denote respectively ,...

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  11. The coefficient of x^(7) in the expansion of (1 - x- x^(2) + x^(3...

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  12. Find the number of ways in which five identical balls can be distribut...

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  13. A committee of 12 is to be formed from nine women and eight men. In ho...

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  14. A committee of 12 is to be formed from nine women and eight men. In ho...

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  15. A question paper consists of two sections having respectively 3 and 4 ...

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  16. Let R =(5 sqrt5+11)^(2n+1) and and f=R-[R], where [ ] denotes the grea...

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  17. Give that : C1+2C2 x+3C3 x^2+.....+2n. C(2n). x^(2n-1)=2n(1+x)^(2n-1)...

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  18. Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C(2r-1) = 0 , wh...

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  19. If a(0),a(1),a(2) ,…, a(2n) are the coefficients in the expansion ...

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

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