Binomial Theorem

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