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Prove that ^nC0 "^(2n)Cn-^nC1 ^(2n-2)Cn ...

Prove that `^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n`

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If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) *""^(2n)C_(n) - C_(1) *""^(2n-2)C_(n) + C_(2) *""^(2n-4) C_(n) -…= 2^(n)

If (18x^2+12x+4)^n = a_0 +a_(1x)+ a_(2x)^2 +......+ a_(2n)x^(2n) , prove that a_r= 2^n3^r ( "^(2n)C_r + "^(n)C_1 "^(2n-2)C_r + "^(n)C_2 "^(2n-4)C_r + ....

Prove that .^n C_0 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2- .. =(-1)^n .

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Prove that (.^(2n)C_0)^2-(.^(2n)C_1)^2+(.^(2n)C_2)^2-..+(.^(2n)C_(2n))^2 = (-1)^n.^(2n)C_n .

show that ^nC_r+ ^(n-1)C_(r-1)+ ^(n-1)C_(r-2)= ^(n+1)C_r

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Prove that (2nC_0)^2-(2nC_1)^2+(2nC_2)^2+.....+(2nC_2n)^2=(-1)^n2nC_n

If (1+x)^(n)=^(n)C_(0)+^(n)C_(1)x+^(n)C_(2)x^(2)+…+^(n)C_(n)x^(n) , prove that, nC_(0) +2^(n)C_(1)+3^(n)C_(2)+…+(n+1)^(n)C_(n)=(n+2)*2^(n-1) .

CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. If every pair from among the equations x^2+a x+b c=0. x^2+b x+c a=0,a ...

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  2. Prove that ^mC1^n Cm-^m C2^(2n)Cm+^m C3^(3n)Cm-.....=(-1)^(m-1)n^mdot

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  3. Prove that ^nC0 "^(2n)Cn-^nC1 ^(2n-2)Cn +^nC2 ^(2n-4)Cn =2^n

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  4. Find the sum sum(r=0) .^(n+r)Cr .

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  5. Find the value of (sumsum)(0leiltjlen) (i+j)(""^(n)C(i)+""^(n)C(j)).

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  6. Find the sum sumsum(0lt=i<jlt=n-1)^n Cidot

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  7. Find the value of underset(0leiltjlen)(sumsum)(.^(n)C(i)+.^(n)C(j)).

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  8. Find the sum (sumsum)(0leiltjlen) ""^(n)C(i).""^(n)C(j).

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  9. Prove that sum(r=0)^ssum(s=1)^n^n Cs^s Cr=3^n-1.

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  10. Find the sum sumsum(0leiltjlen)"^nCi

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  11. Find the coefficient of x^4 in the expansion of (x/2-3/x^2)^10.

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  12. Find the term in(3sqrt(((a)/(sqrt(b))) + (sqrt((b)/ (3sqrt(a))))^(21) ...

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  13. Using the binomial theorem, evaluate (102)^5 .

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  14. Find the 6th term in expansion of (2x^2-1/(3x^2))^(10)dot

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  15. Find a, if 17th and 18th terms in the expansions of (2+a)^50 are equal...

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  16. Find n , if the ratio of the fifth term from the beginning to the ...

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  17. Simplify: x^5+10 x^4a+40 x^3a^2+80 x^2a^3+80 x a^4+32 a^5dot

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  18. Find the value of (18^3+7^3+3.18.7.25) /(3^6+6.243.2+15.81.4+20.27.8+1...

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  19. Find the approximation of (0. 99)^5 using the first three terms of its...

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  20. If for n in N, underset(k=0)overset(2n)sum(-1)^(k)(.^(2n)C(k))^(2) = A...

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