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Prove that .^100 C0^(100)C2+^(100)C2^(10...

Prove that `.^100 C_0^(100)C_2+^(100)C_2^(100)C_4+^(100)C_4^(100)C_6++^(100)C_(98)^(100)C_(100)=1/2[.^(200)C_(98)-^(100)C_(49)]dot`

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The sum of the series (.^(101)C_(1))/(.^(101)C_(0)) + (2..^(101)C_(2))/(.^(101)C_(1)) + (3..^(101)C_(3))/(.^(101)C_(2)) + "….." + (101..^(101)C_(101))/(.^(101)C_(100)) is "____" .

The total number of different terms in the product (.^(101)C_(0) - .^(101)C_(1)x+.^(101)C_(2)x^(2)-"….."-.^(101)C_(101)x^(101))(1+x+x^(2)+"…."+x^(100))^(101) is "____" .

Let t_(100)=sum_(r=0)^(100)(1)/(("^(100)C_(r ))^(5)) and S_(100)=sum_(r=0)^(100)(r )/(("^(100)C_(r ))^(5)) , then the value of (100t_(100))/(S_(100)) is (a) 1 (b) 2 (c) 3 (d) 4

Prove that int_0^(102)(x-1)(x-2)..(x-100)xx(1/(x-1)+1/(x-2) +.... .+ 1/(x-100))dx=101 !-100 !

An equation a_(0)+a_(1)x+a_(2)x^(2)+....+a_(99)x^(99)+x^(100)=0" has roots "^(99)C_(0),^(99)C_(1),^(99)C_(2),...,^(99)C_(99) The value of a_(98) is -

An equation a_(0)+a_(1)x+a_(2)x^(2)+....+a_(99)x^(99)+x^(100)=0" has roots "^(99)C_(0),^(99)C_(1),^(99)C_(2),...,^(99)C_(99) The value of a_(99) is equal to -

If A=[{:(1,0,0),(1,0,1),(0,1,0):}] , then which is true (a) A^(3)-A^(2)=A-I (b) det. (A^(100)-I)=0 (c) A^(200)=[(1,0,0),(100,1,0),(100,0,1)] (d) A^(100)=[(1,1,0),(50,1,0),(50,0,1)]

Let P =sum_(r=1)^(50)(""^(50+r)C_(r)(2r-1))/(""^(50)C_(r)(50+r)), R = sum_(r=0)^(100)(-1)^(r) (""^(100)C_(r))^(2) The value of P - R is equal to

The value of ((100),(0))((200),(150))+((100),(1))((200),(151))+......+((100),(50))((200),(200)) equals (where ((n),(r ))="^(n)C_(r) )

CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. If n=12 m(m in N), prove that .^n C0-(.^n C2)/((2+sqrt(3))^2)+(.^n C...

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  2. Prove that in the expansion of (1+x)^(n) (1+y)^(n) (1+z)^(n), the sum ...

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  3. Prove that .^100 C0^(100)C2+^(100)C2^(100)C4+^(100)C4^(100)C6++^(100)C...

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  4. Prove that sum(r=1)^(m-1)(2r^2-r(m-2)+1)/((m-r)^m Cr)=m-1/mdot

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  5. Find the coefficients of x^(50) in the expression (1+x)^(1000)+x(1+x)^...

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  6. If a, b, c are in GP and a, x, b, y are in AP Then prove that, a/x + ...

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  7. If .^(n+1)C(r+1):^n Cr:^(n-1)C(r-1)=11 :6:3, then n r= 20 b. 30 c. 40...

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  8. If the last tem in the binomial expansion of (2^(1/3)-1/(sqrt(2)))^n i...

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  9. Find the last two digits of the number (23)^(14)dot

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  10. The value of x for which the sixth term in the expansion of [2^(log(...

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  11. If the 6th term in the expansion of (1/x^(8//3)+x^2log(10)x)^8 is 5600...

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  12. The total number of terms which are dependent on the value of x in the...

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  13. In the expansion of (3^(-x//4)+3^(5x//4))^(n) the sum of binomial coef...

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  14. If n is an integer between 0 and 21, then the minimum value of n!(21-n...

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  15. If R is remainder when 6^(83)+8^(83) is divided by 49, then find the v...

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  16. Let a and b be the coefficient of x^(3) in (1+x+2x^(2) + 3x^(3))^(4) a...

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  17. Let 1+ underset(r=1)overset(10)sum(3^(r)..^(10)C(r)+r..^(10)C(r)) = 2^...

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  18. Let a=3^(1//224)+1 and for all n ge 3, let f(n)=""^(n)C(0)a^(n-1)-""...

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  19. If the constant term in the binomial expansion of (x^2-1/x)^n ,n in N...

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  20. The largest value of x for which the fourth tem in the expansion (5^(...

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