Prove that the the coefficient of $x^{5}$ in (1 + x + $x^{3})^{9}$ is 378.

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Solution

${(1 + x) + x^{3}}^{9}$
$= {1 + x}^{9} +^{9} C_{1} {1 + x}^{8} . (x^{3}) + . . . . . .$
Other forms will contain powers of x greater than 5
\therefore Co-eficient of $x^{5} i n^{9} C_{5} + 9 .^{8} C_{2}$
$= \frac{9.8.7 . 6. 5}{1. 2. 3. 4. 5} + 9. \frac{8.7}{1.2} = 126 + 252 = 378$

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