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Statement - 1 sum(r=0)^(n) (r + 1) ""^(...

Statement - 1 ` sum_(r=0)^(n) (r + 1) ""^(n)C_(r) = (n+2)*2^(n-1)`
Statement-2 ` sum_(r=0)^(n) (r +1)""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-1)` .

A

Statement 1 is false, statement 2 is true

B

Statement 1 is true, statement 2 is true, Statement 2 is a correct explanation of

C

Statement 1 is true, statement 2 is true, Statement 2 is not a correct explanation of Statement 1

D

Statement 1 is true, statement 2 is false

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The correct Answer is:
B
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Stetemet - 1: sum_(r=0)^(n) r. ""^(n)C_(r) = n 2^(n-1) Statement-2: sum_(r=0)^(n) r. ""^(n)C_(r) x^(r) = n (1 + x )^(n-1) x

Statement-1 sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = nx (1 - x)^(n -1) Statement-2: sum_(r=0)^(n)r ""^(n)C_(r) x^(r) (-1)^(r) =0

Statement -1: sum_(r=0)^(n) r(""^(n)C_(r))^(2) = n (""^(2n -1)C_(n-1)) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))^(2)= ""^(2n)C_(n)

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}

Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

Statement-1 : sum_(r=0)^(n) r^(2) ""^(n)C_(r) x^(r) = n (n-1) x^(2) (1 + x)^(n-2) + nx (1 +x)^(n-1) Statement-2: sum_(r=0)^(n) r^(2) ""^(n)C_(r) = n (n-1)2^(n-2)+ n2^(n-1) .

If x + y = 1 , prove that sum_(r=0)^(n) r""^(n)C_(r) x^(r ) y^(n-r) = nx .

If a_(n) = sum_(r=0)^(n) (1)/(""^(n)C_(r)) , find the value of sum_(r=0)^(n) (r)/(""^(n)C_(r))

Prove that sum_(r=0)^(n) ""^(n)C_(r )sin rx. cos (n-r)x = 2^(n-1) xx sin nx .

sum_(r=0)^(n)((r+2)/(r+1))*""^(n)C_(r) is equal to :

VMC MODULES ENGLISH-BINOMIAL THEOREM-LEVEL 1
  1. sum(r=0)^(n-1) (""^(n)C(r))/(""^(n)C(r) + ""^(n)C(r+1)) is equal to

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  2. Find the coefficient of x^7 in the expansion of (1 - x -x^2 + x^3)^(6)...

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  3. Statement - 1 sum(r=0)^(n) (r + 1) ""^(n)C(r) = (n+2)*2^(n-1) Stat...

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  4. if n>=2, ""^(n+1)C(2)+2(""^(2)C(2)+""^(3)C(2)+""^(4)C(2)+....+""^(n)...

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  5. In the coefficients of rth, (r+1)t h ,a n d(r+2)t h terms in the binom...

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  6. If s(n)=sum(r=0)^(n)(1)/(.^(n)C(r))and t(n)=sum(r=0)^(n)(r)/(.^(n)C(r)...

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  7. The sum ""^(20)C(0)+""^(20)C(1)+""^(20)C(2)+……+""^(20)C(10) is equal t...

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  8. Find the coefficient of x^5 in (1+2x+3x^2...........)^(-3/2)

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  9. The coefficient of x^2y^3 in the expansion of (1-x+y)^(20) is (20 !)/(...

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  10. The coefficient of x^(10) in the expansion of (1+x^2-x^3)^8 is 476 b. ...

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  11. The reamainder left out when 8^(2n) - (62)^(2n+1) is divided by 9 is

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  12. If ak is the coefficient of x^k in the expansion of (1+x+x^2)^n for k ...

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  13. The coefficient of x^(28) in the expansion of (1+x^3-x^6)^(30) is 1 b....

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  14. The coefficient of x^(n) in the polynomial (x+.^(n)C(0))(x+3.^(n)C(1))...

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  15. If (1+2x+x^2)^n=sum(r=0)^(2n)ar x^r ,then ar is a.(.^nC2)^2 b. .^n ...

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  16. The value of sum(r=0)^nsum(s=0)^nsum(t=0)^nsum(u=0)^n(1) is

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  17. If n is an even natural number , find the value of sum(r=0)^(n) ((...

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  18. sum(r=0)^n(-2)^r*(nCr)/((r+2)Cr) is equal to

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  19. If (1+2x+3x^2)^(10)=a0+a1x+a2x^2++a(20)x^(20),t h e na1 equals 10 b. 2...

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  20. The remainder, if 1+2+2^2++2^(1999) is divided by 5 is.

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