Prove that `^10 C_1(x-1)^2-^(10)C_2(x-2)^2+^(10)C_3(x-3)^2+-^(10)C_(10)(x-10)^2=x^2`

Let X=(\ ^(10)C_1)^2+2(\ ^(10)C_2)^2+3(\ ^(10)C_3)^2+\ ddot\ +10(\ ^(10)C_(10))^2 , where \ ^(10)C_r , r in {1,\ 2,\ ddot,\ 10} denote binomial coefficients. Then, the value of 1/(1430)\ X is _________.

4(10/x)^2-6(10/x)^2+3(10/x)^2=1

The value of (.^(21)C_(1) - .^(10)C_(1)) + (.^(21)C_(2) - .^(10)C_(2)) + (.^(21)C_(3) - .^(10)C_(3)) + (.^(21)C_(4) - .^(10)C_(4)) + … + (.^(21)C_(10) - .^(10)C_(10)) is

Find the value of (.^(10)C_(10))+(.^(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)) .

In the expansion off (1+x)^(10)=.^(10)C_(0)+.^(10)C_(1)x+.^(10)C_(2)x^(2)+ . . .+.^(10)C_(10)x^(10) , then value of 528[(.^(10)C_(0))/(2)-(.^(10)C_(1))/(3)+(.^(10)C_(2))/(4)-(.^(10)C_(3))/(5)+ . . .+(.^(10)C_(10))/(12)] is equal to________.

"^10(C_0)^2 - "^10(C_1)^2 + "^10(C_2)^2 - ...... - ( "^10C_9)^2 + ( "^10C_10)^2=

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) - C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

The solution set "^10C_(x-1)>2 . "^10C_x is

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) .

Find the sum 2..^(10)C_(0) + (2^(2))/(2).^(10)C_(1) + (2^(3))/(3).^(10)C_(2)+(2^(4))/(4).^(10)C_(3)+"...."+(2^(11))/(11).^(10)C_(10) .