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Binomial Theorem

JEE Main 2026 (Online) 22nd January Morning ShiftMCQHard📈 Topic in 43% papers+4−1
Ques. 1JEE Main 2026, 22 January (Morning Shift)

The coefficient of x48 in (1 + x) + 2(1 + x)2 + 3(1 + x)3 + + 100(1 + x)100 is equal to

  1. 100 100C49 - 100C48
  2. 100 101C49 - 101C50
  3. 100C50 + 101C49
  4. 100 100C49 - 100C50

Answer

(2) 100 101C49 - 101C50

Solution

Let y = 1 + x and S = y + 2y2 + 3y3 + + 100y100 (an A.G.P.), so

yS = y2 + 2y3 + + 99y100 + 100y101

Subtracting,

S(1 - y) = y + y2 + + y100 - 100y101

S(1 - y) = y101 - yy - 1 - 100y101

1 - y = -x and y - 1 = x,

-xS = y101 - yx - 100y101

S = 100(1 + x)101x - (1 + x)101x2 + 1 + xx2

The last term gives only x-2 and x-1, so the coefficient of x48 in S is

100 (coefficient of x49 in (1 + x)101) - (coefficient of x50 in (1 + x)101)

= 100 101C49 - 101C50

Answer: (2)

Keep practising — Binomial Theorem

Ques. 2JEE Main 2026, 22 January (Evening Shift)

Let Cr denote the coefficient of xr in the binomial expansion of (1 + x)n, n N, 0 r n. If Pn = C0 - C1 + 223C2 - 234C3 + + (-2)nn+1Cn, then the value of n=125 1P2n equals

  1. 675
  2. 580
  3. 525
  4. 650
Ques. 3JEE Main 2026, 21 January (Morning Shift)

If the coefficient of x in the expansion of (ax2 + bx + c)(1 - 2x)26 is -56 and the coefficients of x2 and x3 are both zero, then a + b + c is equal to :

  1. 1483
  2. 1300
  3. 1500
  4. 1403
Ques. 4JEE Main 2026, 21 January (Evening Shift)

If ( 115C0 + 115C1 ) ( 115C1 + 115C2 ) ( 115C12 + 115C13 ) = 1314C0 14C1 14C12, then 30 is equal to .

Ques. 5JEE Main 2025, 8 April (Evening Shift)

The number of integral terms in the expansion of (5 12 + 7 18 )1016 is:

  1. 127
  2. 128
  3. 130
  4. 129
Ques. 6JEE Main 2025, 8 April (Evening Shift)

The product of the last two digits of (1919)1919 is

Ques. 7JEE Main 2025, 7 April (Morning Shift)

The remainder when ((64)(64) )(64) is divided by 7 is equal to

  1. 4
  2. 6
  3. 3
  4. 1
Ques. 8JEE Main 2025, 7 April (Evening Shift)

The sum of the series 2 1 20C4 - 3 2 20C5 + 4 3 20C6 - 5 4 20C7 + + 18 17 20C20, is equal to .

Ques. 9JEE Main 2025, 4 April (Morning Shift)

For an integer n 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)2n - 3 is 16, then the distance of the point P (2n - 1, n2 - 4n ) from the line x + y = 8 is

  1. 2
  2. 2 2
  3. 5 2
  4. 3 2
Ques. 10JEE Main 2025, 4 April (Morning Shift)

In the expansion of ( [3]2 + 1 [3]3 )n, n N, if the ratio of 15th term from the beginning to the 15th term from the end is 16, then the value of nC3 is

  1. 4960
  2. 2300
  3. 1040
  4. 4060
Ques. 11JEE Main 2025, 4 April (Evening Shift)

If 12 (15C1 ) + 22 (15C2 ) + 32 (15C3 ) + + 152 (15C15 ) = 2m 3n 5k, where m, n, k N, then m + n + k is equal to :

  1. 20
  2. 19
  3. 18
  4. 21

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All JEE Main Mathematics chapters
Class 11SetsRelations and FunctionsTrigonometric FunctionsComplex Numbers and Quadratic EquationsPermutations and CombinationsBinomial TheoremSequences and SeriesStraight LinesConic SectionsIntroduction to Three Dimensional GeometryLimits and DerivativesStatisticsProbability
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