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Prove that ^10 C1(x-1)^2-^(10)C2(x-2)^2+...

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`

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Let X=(^(10)C_(1))^(2)+2(^(10)C_(2))^(2)+3(^(10)C_(3))^(2)+...+10(^(10)C_(10))^(2) where ^(10)C_(r),r in{1,2,;10} denote binomial coefficients.Then,the value of (1)/(1430)X is

^10(C_(0))^(2)-^(10)(C_(1))^(2)+^(10)(C_(2))^(2)-......-(^(10)C_(9))^(2)+(^(10)C_(10))^(2)=

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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________.

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Evaluate ""^(10)C_1 + ""^(10)C_2 + ""^(10)C_3 + ………+""^10C_10