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The smallest positive integers is . . . ...

The smallest positive integers is . . . . . . . . . . . . . . . . . .

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Let m be the smallest positive integer such that the coefficient of x^2 in the expansion of (1+x)^2 + (1 +x)^3 + (1 + x)^4 +........+ (1+x)^49 + (1 + mx)^50 is (3n + 1) .^51C_3 for some positive integer n. Then the value of n is