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In the polynomial (x+C(n, 0))(x+3C(n, 1))(x+5C(n, 2))⋯ l(x+(2n+1)C(n, n) r), what is the coefficient of xⁿ?

1. n· 2ⁿ
2. n· 2ⁿ⁺¹
3. (n+1)· 2ⁿ
4. n· 2ⁿ⁺¹

Correct answer: (n+1)· 2ⁿ

Solution

The coefficient of x^n is the sum of the constant terms: sum_{k=0}^{n} (2k+1)C(n,k) = 2*sum k*C(n,k) + sum C(n,k) = 2*n*2^(n-1) + 2^n = n*2^n + 2^n = (n+1)*2^n.

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