Binomial Theorem questions
776 published questions with worked solutions.
- Let m,n N and (2,n)=1. If 30 30 0+29 30 1+ +2 30 28 +1 30 29 =n 2^m, then n+m is equal to (Here nk= ^nC_k)
- Let n be a positive integer. Let A= _ k=0 ^n(-1)^k , ^nC_k [ ( 12 )^k+ ( 34 )^k+ ( 78 )^k+ ( 15 16 )^k+ ( 31 32 )^k ]. If 63A=1-...
- Let the coefficients of third, fourth and fifth terms in the expansion of (x+ a x^2 )^n, x 0, be in the ratio 12:8:3. Then the...
- Let ^nC_r denote the binomial coefficient of x^r in the expansion of (1+x)^n. If _ k=0 ^ 10 (2^2+3k ) ^ 10 C_k= 3^ 10 + 2^ 10 , ,...
- The ratio of the coefficient of the middle term in the expansion of (1+x)^ 20 and the sum of the coefficients of two middle terms...
- Let nk denote ^nC_k and [ n k ]= cases nk,& if 0 k n 0,& otherwise. cases If A_k= _ i=0 ^9 9i [ 12 12-k+i ]+ _ i=0 ^8 8i [ 13...
- If 3^6 4^4 k is the term, independent of x, in the binomial expansion of ( x4- 12 x^2 )^ 12 , then k is equal to
- If the coefficient of a^7b^8 in the expansion of (a+2b+4ab)^ 10 is K 2^ 16 , then K is equal to
- Let n N and [x] denote the greatest integer less than or equal to x. If the sum of (n+1) terms ^nC_0,3 ^nC_1,5 ^nC_2,7 ^nC_3, is...
- Let C_r denote the binomial coefficient of x^r in the expansion of (1+x)^ 10 . If for , R, C_1+3 2 ,C_2+5 3 ,C_3+ upto 10 terms =...
- If the sum of the co-efficients of all the positive even powers of x in the binomial expansion of (2x^3+ 3x )^ 10 is 5^ 10 - 3^9,...
- If ( ^ 40 C_0 )+ ( ^ 41 C_1 )+ ( ^ 42 C_2 )+ + ( ^ 60 C_ 20 )= mn , ^ 60 C_ 20 and m and n are coprime, then m+n is equal to
- If the coefficient of x^ 10 in the binomial expansion of ( x 5^ 1/4 + 5 x^ 1/3 )^ 60 is 5^k l, where l,k N and l is co-prime to...
- If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of (x^n+ 2 x^5 )^7 is 939, then the sum...
- The number of positive integers k such that the constant term in the binomial expansion of (2x^3+ 3 x^k )^ 12 , x 0 is 2^8 ,...
- Let the coefficients of x^ -1 and x^ -3 in the expansion of (2x^ 15 - 1 x^ 15 )^ 15 , x>0, be m and n respectively. If r is a...
- If the maximum value of the term independent of t in the expansion of (t^2x^ 15 + (1-x)^ 1 10 t )^ 15 , x 0 is K, then 8K is...
- If the coefficients of x and x^2 in the expansion of (1+x)^p(1-x)^q, p,q 15, are -3 and -5 respectively, then the coefficient of...
- Let for the 9^ th term in the binomial expansion of (3+6x)^n, in the increasing powers of 6x, to be the greatest for x= 32, the...
- Let the coefficients of the middle terms in the expansion of ( 1 6 + x )^4, (1-3 x)^2 and (1- 2x )^6, >0, respectively form the...
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