Binomial Theorem

If 26 ( 233 (12C2 )+ 255 (12C4 )+ 277 (12C6 ) . .+ + 21313 (12C12 ) )=313- , then is equal to :

1. 45
2. 48
3. 51
4. 54

If the coefficients of the middle terms in the binomial expansions of (1+ x)26 and (1- x)28, 0, are equal, then the value of is :

1. 1
2. 1413
3. 277
4. 727

If (1-x3 )10= r=010arxr(1-x)30-2r, then 9a9a10 is equal to .

If the sum of the coefficients of x7 and x14 in the expansion of ( 1x3-x4 )n, x 0, is zero, then the value of n is .

The coefficient of x2 in the expansion of (2x2+ 1x )10, x 0, is :

1. 3240
2. 3360
3. 3480
4. 3600

Let the smallest value of k N, for which the coefficient of x3 in (1+x)3+(1+x)4+(1+x)5+ +(1+x)99+(1+kx)100, x 0, is (43n+ 1014 ) (100C3 ) for some n N, be p. Then the value of p+n is :

1. 10
2. 11
3. 12
4. 13

In the expansion of (9x- 13 x )18, x>0, if the term independent of x is (221)k, then k is equal to :

1. 84
2. 78
3. 168
4. 198

The number of elements in the set S= \(r,k) : k Z and 36Cr+1= 6 (35Cr ) (k2-3 ) \, is :

1. 2
2. 4
3. 8
4. 16

If for 3 r 30, (30C30-r )+3 (30C31-r )+3 (30C32-r )+ (30C33-r )=mCr, then m equals :

1. 31
2. 32
3. 33
4. 34

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are false
3. Both Statement I and Statement II are true
4. Statement I is true but Statement II is false

The sum of the coefficients of x499 and x500 in (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + + x1000 is :

1. 1002C501
2. 1001C501
3. 1000C501
4. 1002C500

Let S = 125! + 13! 23! + 15! 21! + up to 13 terms. If 13 S = 2kn!, k N, then n + k is equal to

1. 50
2. 52
3. 49
4. 51

The sum of all possible values of n N, so that the coefficients of x, x2 and x3 in the expansion of (1 + x2)2(1 + x)n, are in arithmetic progression is :

1. 12
2. 9
3. 3
4. 7

The value of 100C5051 + 100C5152 + + 100C100101 is:

1. 2101101
2. 2100101
3. 2100100
4. 2101100

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

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

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

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

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

1. 127
2. 128
3. 130
4. 129

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

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

1. 4
2. 6
3. 3
4. 1

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

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

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

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

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

1. 16923
2. 18817
3. 3763
4. 33845

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

1. 27
2. 81
3. 18
4. 9

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

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

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

1. 11
2. 20
3. 24
4. 15