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Coefficient of 28th-Power in Expansion

Source: AMC 12 2008A Problem 19

February 17, 2008
functionbinomial coefficientsAMC

Problem Statement

In the expansion of \left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{27}\right)\left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{14}\right)^2, what is the coefficient of x28 x^{28}? <spanclass=latexbold>(A)</span> 195<spanclass=latexbold>(B)</span> 196<spanclass=latexbold>(C)</span> 224<spanclass=latexbold>(D)</span> 378<spanclass=latexbold>(E)</span> 405 <span class='latex-bold'>(A)</span>\ 195 \qquad <span class='latex-bold'>(B)</span>\ 196 \qquad <span class='latex-bold'>(C)</span>\ 224 \qquad <span class='latex-bold'>(D)</span>\ 378 \qquad <span class='latex-bold'>(E)</span>\ 405