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Binomial Theorem · Question 16

JEE Main 2026 (Online) 22nd January Evening ShiftMCQMedium📈 Topic in 52% papers+4−1
1

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

Answer

(1) 675

Solution

General term: Pn = r=0n (-2)rr+1Cr

Crr+1 = 1n+1 n+1Cr+1

Pn = 1n+1 r=0n n+1Cr+1(-2)r = 1-2(n+1) k=1n+1 n+1Ck(-2)k

= (1 - 2)n+1 - 1-2(n+1) = 1 - (-1)n+12(n+1)

For 2n in place of n: P2n = 1 + 12(2n+1) = 12n+1

n=125 1P2n = n=125(2n + 1) = 2 25 262 + 25

= 650 + 25 = 675

Answer: (1)

JEE Main 2026 (Online) 22nd January Evening Shift

Keep practising — Binomial Theorem

2

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
JEE Main 2026 (Online) 21st January Morning Shift
3

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

JEE Main 2026 (Online) 21st January Evening Shift
4

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

  1. 127
  2. 128
  3. 130
  4. 129
JEE Main 2025 (Online) 8th April Evening Shift
5

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

JEE Main 2025 (Online) 8th April Evening Shift
6

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

  1. 4
  2. 6
  3. 3
  4. 1
JEE Main 2025 (Online) 7th April Morning Shift
7

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

JEE Main 2025 (Online) 7th April Evening Shift
8

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
JEE Main 2025 (Online) 4th April Morning Shift
9

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
JEE Main 2025 (Online) 4th April Morning Shift
10

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
JEE Main 2025 (Online) 4th April Evening Shift
11

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

  1. 16923
  2. 18817
  3. 3763
  4. 33845
JEE Main 2025 (Online) 3rd April Morning Shift

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