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Q.
The coefficient of $x^5 y^2 z^2$ in the expansion of $(x+y+2 z)^9$, is
Binomial Theorem
Solution:
$T _{ r +1} \text { in }( x +( y +2 z ))^9 \text { is given by } $
$T _{ r +1}={ }^9 C _{ r } x ^{9- r }( y +2 z )^{ r } $
$\text { Put } r =4, T _5={ }^9 C _4 x ^5( y +2 z )^4={ }^9 C _4 x ^5\left[{ }^4 C _{ p } y ^{4- p }(2 z )^{ p }\right]$
$\text { Put } p =2, T _5={ }^9 C _4{ }^4 C _2 x ^5 y ^2 2^2 z ^2$
Hence coefficient of $x^5 y^4 z^2={ }^9 C_4 \cdot{ }^4 C_2 \cdot 2^2=\frac{9 !}{4 ! 5 !} \cdot \frac{4 !}{2 ! 2 !} 2^2=9 \cdot 8 \cdot 7 \cdot 6=3024$