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Binomial Theorem

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  1. 1.If 26(2^33(^12C_2)+2^55(^12C_4)+2^77(^12C_6)+ +2^1313(^12C_12))=3^13- , then is equal to :
  2. 2.If the coefficients of the middle terms in the binomial expansions of (1+ x)^26 and (1- x)…
  3. 3.The coefficient of x^2 in the expansion of (2x^2+(1)/(x))^10,x 0, is :
  4. 4.In the expansion of (9x-13x)^18,x>0, if the term independent of x is (221)k, then k is equ…
  5. 5.Let the smallest value of k N, for which the coefficient of x^3 in (1+x)^3+(1+x)^4+(1+x)^5…
  6. 6.If for 3 r 30, (^30C_30-r)+3(^30C_31-r)+3(^30C_32-r)+(^30C_33-r)=^mC_r, then m equals :
  7. 7.Given below are two statements :Statement I :25^13 + 20^13 + 8^13 + 3^13 is divisible by 7…
  8. 8.The sum of the coefficients of x^499 and x^500 in (1 + x)^1000 + x(1 + x)^999 + x^2(1 + x)…
  9. 9.Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+ up to 13 terms.
  10. 10.The sum of all possible values of n N, so that the coefficients of x,x^2 and x^3 in the ex…
  11. 11.The value of ^100C_5051+^100C_5152+ .+^100C_100101 is:
  12. 12.Let C_r denote the coefficient of x^r in the binomial expansion of (1+x)^n,n N,0 r n.
  13. 13.The coefficient of x^48 in (1+x)+2(1+x)^2+3(1+x)^3++100(1+x)^100 is equal to
  14. 14.If the coefficient of x in the expansion of (ax^2+bx+c)(1-2x)^26 is -56 and the coefficien…
  15. 15.The number of integral terms in the expansion of (5^(1)/(2)+7^(1)/(8))^1016 is:
  16. 16.The remainder when ((64)^(64))^(64) is divided by 7 is equal to
  17. 17.If 1^2 (^15C_1)+2^2 (^15C_2)+3^2 (^15C_3)+ +15^2 (^15C_15)=2^m 3^n 5^k, where m,n,k N, the…
  18. 18.For an integer n 2, if the arithmetic mean of all coefficients in the binomial expansion o…
  19. 19.In the expansion of (√32^+)n,n N, if the ratio of 15^ term from the beginning to the 15^ t…
  20. 20.The sum of all rational terms in the expansion of (2+√3)^8 is :
  21. 21.If _r=1^9(r+32^r) ^9C_r= ((3)/(2))^9- , , N, then ( + )^2 is equal to
  22. 22.If, _r=0^10(10^r+1-110^r).^11C_r+1=_ 11-11^1110^10,, then, , , , is, , equal, , to:
  23. 23.The largest n N such that 3^n divides 50 !
  24. 24.The term independent of x in the expansion of ((x+1)(x^2/3+1-x^1/3)-(x-1)(x-x^1/2))^10,x>1…
  25. 25.The remainder, when 7^103 is divided by 23, is equal to:
  26. 26.The least value of n for which the number of integral terms in the Binomial expansion of (…
  27. 27.Let the coefficients of three consecutive terms T_r, T_r+1 and T_r+2 in the binomial expan…
  28. 28.Suppose A and B are the coefficients of 30^ and 12^ terms respectively in the binomial exp…
  29. 29.For some n 10, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expan…
  30. 30.If in the expansion of (1+x)^p(1-x)^q, the coefficients of x and x^2 are 1 and -2 , respec…
  31. 31.Let , , and be the coefficients of x^7,x^5,x^3 and x respectively in the expansion of gath…
  32. 32.The sum of the coefficient of x^2/3 and x^-2/5 in the binomial expansion of (x^2/3+(1)/(2)…
  33. 33.The coefficient of x^70 in x^2(1+x)^98+x^3(1+x)^97+x^4(1+x)^96+ +x^54(1+x)^46 is ^99C_p-^4…
  34. 34.If the term independent of x in the expansion of (ax^2+12x^3)^10 is 105 , then a^2 is equa…
  35. 35.If the constant term in the expansion of ((√53)/()x^+)12,x 0, is 2^8 √53, then 25 is equal…
  36. 36.If the coefficients of x^4, x^5 and x^6 in the expansion of (1+x)^n are in the arithmetic …
  37. 37.The sum of all rational terms in the expansion of (2^(1)/(5)+5^(1)/(3))^15 is equal to :
  38. 38.Let m and n be the coefficients of seventh and thirteenth terms respectively in the expans…
  39. 39.Let a be the sum of all coefficients in the expansion of (1-2x+2x^2)^2023(3-4x^2+2x^3)^202…
  40. 40.^n-1C_r=(k^2-8)^nC_r+1 if and only if :
  41. 41.If A denotes the sum of all the coefficients in the expansion of (1-3x+10x^2)^n and B deno…
  42. 42.Let (a+bx+cx^2)^10= _i=0^20p_ix^i,a,b,c N.
  43. 43.The coefficient of x^5 in the expansion of (2x^3-13x^2)^5 is :
  44. 44.Fractional part of the number [4^202215] is equal to
  45. 45.If (1)/(n+1)^nC_n+(1)/(n)^nC_n-1+ +(1)/(2)^nC_1+^nC_0=(1023)/(10) then n is equal to :
  46. 46.The sum, of the coefficients of the first 50 terms in the binomial expansion of (1-x)^100,…
  47. 47.The sum of the coefficients of three consecutive terms in the binomial expansion of [(1+x)…
  48. 48.If the 1011^ term from the end in the binominal expansion of ((4x)/(5)-(5)/(2x))^2022 is 1…
  49. 49.Let the number (22)^2022+(2022)^22 leave the remainder when divided by 3 and when divided …
  50. 50.If the coefficients of x and x^2 in (1+x)^p(1-x)^q are 4 and -5 respectively, then 2p+3q i…