Find the value of `1/(81^n)-(10)/(81^n)^(2n)C_1+(10^2)/(81^n)^(2n)C_2-(10^3)/(81^n)^(2n)C_3++(10^(2n))/(81^n)` .

Find the value of 1/(81^n)-(10)/(81^n)C_1+(10^2)/(81^n)C_2-(10^3)/(81^n)C_3++(10^(2n))/(81^n) .

Find the value of n: (i) .^(n)C_(10)=^(n)C_(16) (ii) .^(15)C_(n) =^(15)C_(n+3) (iii) .^(10)C_(n) =^(10)C_(n+2) (iv) .^(25)C_(3n) =.^(25)C_(n+1) (v) .^(n)C_(r) =.^(n)C_(r-2)

Find .^(n)C_(1)-(1)/(2).^(n)C_(2)+(1)/(3).^(n)C_(3)- . . . +(-1)^(n-1)(1)/(n).^(n)C_(n)

The value of |1 1 1^n C_1^(n+2)C_1^(n+4)C_1^n C_2^(n+2)C_2^(n+4)C_2| is

The value of ""^(n)C_(n)+""^(n+1)C_(n)+""^(n+2)C_(n)+….+""^(n+k)C_(n) :

Prove that (^(2n)C_0)^2-(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

Each question has four choices a, b, c and d, out of which only one is correct. Each question contains STATEMENT 1 and STATEMENT 2. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT1. Both the statements are TRUE but STATEMENT 2 is NOT the correct explanation of STATEMENT 1. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE. Statement 1: The value of (^(10)^C_0)+(^(10)C_0+(10)C_1)+(^(10)C_0+(10)C_1+(10)C_2)++(^(10)C_0+(10)C_1+(10)C_2++(10)C_9) is 10 2^9 . Statement 2: ^n C_1+2^n C_2+3^n C_3+ n^n C_n=n2^(n-1) .

if (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a++.^(n)C_(2)a^(2)+ . . .+.^(n)C_(n)a^(n) , then prove that (.^(n)C_(1))/(.^(n)C_(0))+(2(.^(n)C_(2)))/(.^(n)C_(1))+(3(.^(n)C_(3)))/(.^(n)C_(2))+. . . +(n(.^(n)C_(n)))/(.^(n)C_(n-1))= Sum of first n natural numbers.

The value of ""(n)C_(1). X(1 - x )^(n-1) + 2 . ""^(n)C_(2) x^(2) (1 - x)^(n-2) + 3. ""^(n)C_(3) x^(3) (1 - x)^(n-3) + ….+ n ""^(n)C_(n) x^(n) , n in N is

Evaluate Lt_(n to oo)[(1)/(1+n)+(1)/(2+n)+(1)/(3+n)+"...."+(1)/(10n)]