The coefficient of x5 in the expansion of \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
Options
51C5
9C5
31C6 − 21C6
30C5 + 20C5
Solution
31C6 − 21C6
\[\text{ We have } \left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . \left( 1 + x \right)^{30} \]
\[ = \left( 1 + x \right)^{21} \left[ \frac{\left( 1 + x \right)^{10} - 1}{\left( 1 + x \right) - 1} \right]\]
\[ = \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]\]
\[\text{ Coefficient of } x^5 \text{ in the given expansion = Coefficient of } x^5 \text{ in } \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]\]
\[ = \text{ Coefficient of } x^6 \text{ in }\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]\]
\[ =^{31} C_6 -^{21} C_6 \]
