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

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

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

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

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

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 .

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

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

If = 1 + r=16(-3)r-1 12C2r-1, then the distance of the point (12, 3) from the line x - 3y + 1 = 0 is .

If r=05 11C2r+12r+2= mn, (m,n)=1, then m-n is equal to .

If r=130 r2 (30Cr )230Cr-1= 229, then is equal to .

The coefficient of x70 in x2(1 + x)98 + x3(1 + x)97 + x4(1 + x)96 + + x54(1 + x)46 is 99Cp - 46Cq. Then a possible value of p + q is :

1. 61
2. 83
3. 55
4. 68

Let = r=0n (4r2 + 2r + 1 )nCr and = ( r=0n nCrr + 1 ) + 1n + 1. If 140 < 2 < 281, then the value of n is .

If the coefficients of x4, x5 and x6 in the expansion of (1 + x)n are in the arithmetic progression, then the maximum value of n is:

1. 28
2. 21
3. 7
4. 14

Let the coefficient of xr in the expansion of (x+3)n-1+(x+3)n-2(x+2)+(x+3)n-3(x+2)2+ +(x+2)n-1 be r. If r=0n r= n- n, , N, then the value of 2+ 2 equals .

Suppose 2-p, p, 2- , are the coefficients of four consecutive terms in the expansion of (1+x)n. Then the value of p2- 2+6 +2p equals

1. 8
2. 4
3. 6
4. 10

Let = k=0n ( ( n Ck )2k+1 ) and = k=0n-1 ( n Ck n Ck+1k+2 ). If 5 =6 , then n equals .

If 11C12+ 11C23+ + 11C910= nm with (n,m)=1, then n+m is equal to .

If A denotes the sum of all the coefficients in the expansion of (1-3x+10x2 )n and B denotes the sum of all the coefficients in the expansion of (1+x2 )n, then :

1. B=A3
2. 3A=B
3. A=3B
4. A=B3