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Binomial Theorem

JEE Main 2025 (Online) 7th April Evening ShiftNumericalHard📈 Topic in 47% papers+4
Ques. 1JEE 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 .

Answer

34

Solution

The general term (r = 4 to 20) is (-1)r (r - 2)(r - 3) 20Cr.

(r = 4 gives +2 1 20C4, r = 20 gives +18 17 20C20.)

Since (r - 2)(r - 3) is a polynomial in r of degree 2 (< 20),

r=020 (-1)r (r - 2)(r - 3) 20Cr = 0

(because (-1)r nCr = (-1)r r nCr = (-1)r r(r - 1) nCr = 0 for n = 20)

S = - r=03 (-1)r (r - 2)(r - 3) 20Cr

r = 0: (+1)(-2)(-3)(1) = 6

r = 1: (-1)(-1)(-2)(20) = -40

r = 2, 3: the factor (r - 2)(r - 3) = 0

S = -(6 - 40) = 34

Answer: 34

Keep practising — Binomial Theorem

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

The sum of all rational terms in the expansion of (2+ 3)8 is :

  1. 16923
  2. 18817
  3. 3763
  4. 33845
Ques. 6JEE Main 2025, 3 April (Morning Shift)

If r=19 ( r+32r ) 9Cr = ( 32 )9 - , , N, then ( + )2 is equal to

  1. 27
  2. 81
  3. 18
  4. 9
Ques. 7JEE Main 2025, 3 April (Evening Shift)

Let (1 + x + x2 )10 = a0 + a1x + a2x2 + + a20x20. If (a1 + a3 + a5 + + a19) - 11a2 = 121k, then k is equal to .

Ques. 8JEE Main 2025, 2 April (Morning Shift)

The term independent of x in the expansion of ( (x + 1)(x2/3 + 1 - x1/3) - (x - 1)(x - x1/2) )10, x > 1, is :

  1. 240
  2. 120
  3. 150
  4. 210
Ques. 9JEE Main 2025, 2 April (Evening Shift)

If r=010 ( 10r+1 - 110r ) 11Cr+1 = 11 - 11111010, then is equal to :

  1. 11
  2. 20
  3. 24
  4. 15
Ques. 10JEE Main 2025, 29 January (Morning Shift)

The least value of n for which the number of integral terms in the Binomial expansion of ( [3]7 + [12]11)n is 183, is :

  1. 2184
  2. 2172
  3. 2196
  4. 2148
Ques. 11JEE Main 2025, 29 January (Evening Shift)

The remainder, when 7103 is divided by 23, is equal to:

  1. 9
  2. 6
  3. 14
  4. 17

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