Find the 12th term in the expansion (5x+2)^14
I did (14 ncr 11) (5x)^3 (2)^11
364 (125x^3) (2048)
93,184,000x^3. Is that right?
This can be tricky: you can both write $\displaystyle (5x+2)^{14}=\displaystyle{\sum\limits^{14}_{k=0}\b inom{14}{k}(5x)^{14-k}2^k=\sum\limits^{14}_{k=0}\binom{14}{k}(5x)^{k}2 ^{14-k}}$
In the first case you obtain $\displaystyle \displaystyle{\binom{14}{11}(5x)^32^{11}}$, as you did, whereas in the second case you
get $\displaystyle \displaystyle{\binom{14}{11}(5x)^{11}2^3}$ , so what's the correct one?
In my opinion, they both are, and this perhaps is the reason why the usually ask what the
coefficient of some power of x is, not what is this or that term.
Tonio