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Binomial Theorem

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  1. 101.If 3030-r+33031-r+33032-r+3033-r=mr for 3 r30, then m equals :
  2. 102.Let the smallest value of k N, for which the coefficient of x^3 in (1+x)^3+(1+x)^4+(1+x)^5…
  3. 103.The coefficient of x^2 in the expansion of (2x^2+1x)^10, x0, is:
  4. 104.Let (1+x+x^2)^10=a_0+a_1x+a_2x^2++a_20x^20.
  5. 105.Let ,,, be the coefficients of x^7,x^5,x^3,x respectively in the expansion of (x+x^3-1)^5+…
  6. 106.For some n10, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion…
  7. 107.Let the coefficients of three consecutive terms T_r,T_r+1,T_r+2 in the binomial expansion …
  8. 108.If _r=0^10(10r-1-(1)/(10)10r)^11C_r+1=11^1110^10, then is equal to:
  9. 109.If _r=1^9r+3r2^-r^9C_r = (32)^9-, , N, then (+)^2 is equal to:
  10. 110.In the expansion of ([3]2+1[3]3)^n, if the ratio of the 15th term from the beginning to th…
  11. 111.For an integer n2, if the arithmetic mean of all coefficients in the binomial expansion of…
  12. 112.If A denotes the sum of all the coefficients in the expansion of (1-3x+10x^2)^n and B deno…
  13. 113.If the coefficient of x^30 in the expansion of (1+1x)^6(1+x^2)^7(1-x^3)^8, x0, is , then |…
  14. 114.The value _r=0^2222r23r is
  15. 115.Suppose _r=0^2023r^2,^2023C_r = 20232^2022.
  16. 116.If (^30C_1)^2+2(^30C_2)^2+3(^30C_3)^2++30(^30C_30)^2=60!(30!)^2, then is equal to
  17. 117.Let the sum of the coefficients of the first three terms in the expansion of (x-3x^2)^n, x…
  18. 118.If a_r is the coefficient of x^10-r in the Binomial expansion of (1+x)^10, then _r=1^10r^3…
  19. 119.The constant term in the expansion of (2x+1x)^7(3x^2)^? *(five-term multinomial variant)* …
  20. 120._k=0^6^51-kC_3 is equal to
  21. 121.The remainder when (2023)^2023 is divided by 35 is ________.
  22. 122.If the coefficient of x^9 in ( x^3+1 x)^11 and the coefficient of x^-9 in ( x-1 x^3)^11 ar…
  23. 123.Let the coefficients of three consecutive terms in the binomial expansion of (1+2x)^n be i…
  24. 124.Let K be the sum of the coefficients of the odd powers of x in the expansion of (1+x)^99.
  25. 125.If the coefficient of x^15 in the expansion of (ax^3+1bx^1/3)^15 is equal to the coefficie…
  26. 126.The coefficient of x^301 in (1+x)^500+x(1+x)^499+x^2(1+x)^498++x^500 is:
  27. 127.The 50th root of a number x is 12 and 50th root of another number y is 18.
  28. 128.Let >0 be the smallest number such that the expansion of (x^2/3+2x^3)^30 has a term x^-, N…
  29. 129.The remainder on dividing 5^99 by 11 is ________.
  30. 130.The coefficient of x^-6 in the expansion of (4x5+52x^2)^9 is _______ Answer: 5040
  31. 131.If the constant term in the binomial expansion of (x^5/2-4x^)^9 is -84 and the coefficient…
  32. 132.The value of 11!,50!+13!,48!+15!,46!++149!,2!+151!,1! is
  33. 133.The remainder when 19^200+23^200 is divided by 49, is _______.
  34. 134.Let the sixth term in the binomial expansion of (2^_2(10-3^x)+2^(x-2)_23)^m, in the increa…
  35. 135.If the term without x in the expansion of (x^2/3+x^3)^22 is 7315, then || is equal to ____…
  36. 136.If the coefficients of x^7 in (ax^2+12bx)^11 and x^-7 in (ax-13bx^2)^11 are equal, then:
  37. 137.Among the statements: (S1): 2023^2022-1999^2022 is divisible by 8 (S2): 13(13)^n-11n-13 is…
  38. 138.Let [t] denote the greatest integer t.
  39. 139.The absolute difference of the coefficients of x^10 and x^7 in the expansion of (2x^2+12x)…
  40. 140.If the coefficient of x^7 in (ax-1bx^2)^13 and the coefficient of x^-5 in (ax+1bx^2)^13 ar…
  41. 141.The coefficient of x^7 in (1-x+2x^3)^10 is _______.
  42. 142.If the 1011th term from the end in the binomial expansion of (4x5-52x)^2022 is 1024 times …
  43. 143.If 1n+1^nC_n+1n^nC_n-1++12^nC_1+^nC_0=102310, then n is equal to
  44. 144.Let for x R, S_0(x)=x, S_k(x)=C_kx+k_0^x S_k-1(t),dt where C_0=1, C_k=1-_0^1S_k-1(x),dx, k…
  45. 145.The coefficient of x^5 in the expansion of (2x^3-13x^2)^5 is
  46. 146.The remainder when 3^2022 is divided by 5 is:
  47. 147.The remainder on dividing 1+3+3^2+3^3++3^2021 by 50 is__ Answer: 4
  48. 148.The coefficient of x^101 in the expression (5+x)^500+x(5+x)^499+x^2(5+x)^498++x^500, x>0, …
  49. 149.If the sum of the co-efficient of all the positive even powers of x in the binomial expans…
  50. 150.The remainder when (2021)^2023 is divided by 7 is: