Binomial Theore...

 If in the expansion of (1 + x)^m  (1 – x)ⁿ , the co-efficients of x and x²  are 3 and – 6 respectively, then m is [JEE 99,2 ]

For 2 £r £n, 

Find the largest co-efficient in the expansion of (1 + x)ⁿ , given that the sum of co-efficients of the terms in its expansion is 4096.   [REE 2000 (Mains)]

In the binomial expansion of (a - b)ⁿ , n ³5, the sum of the 5th and 6th terms is zero. Then equals. 

Find the coefficient of x⁴⁹  in the polynomial [REE 2001 (Mains), 3]  where C_r  = ⁵⁰ C_r

The sum   , (where   = O if P < q) is maximum when m is [JEE 2002 (Scr.), 3]

 (a) Coefficient of t²⁴  in the expansion of (1 + t² )¹²  (1 + t¹² ) (1 + t²⁴ ) is [JEE 2003 (Scr.), 3]

(b) Prove that :

     [JEE 2003 (Mains),2]

^n_1 C_r  = (k²  – 3). ⁿ C_{r + 1}^,  if k Π  [JEE 2004 (Scr.)]

 ^n-1 C_r  = (k² - 3). ⁿ C_{r + 1} [JEE 2004 (Scr.)]

The value of

 is, where    = ⁿ C_r       [JEE 2005 (Scr.)]

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is        [JEE 2009]

 For r = 0, 1, ...., 10 let A_r , B_r , C_r  denote, respectively, the coefficient of x^r  in the expansions of (1 + x)¹⁰ , (1 + x)²⁰  and (1 + x)³⁰ . Then   is equal to    [JEE 2010]

Let a_n  denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_n  = the number of such n-digit integers ending with digit 1 and c_n  = the number of such n-digit integers ending with digit 0.     [JEE 2012]

Which of the following is correct ?

Let a_n  denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_n  = the number of such n-digit integers ending with digit 1 and c_n  = the number of such n-digit integers ending with digit 0.         [JEE 2012]

The value of b₆  is