If the sum of the co-efficient of all the positive even powers of x in the binomial expans…

Question

If the sum of the co-efficient of all the positive even powers of $x$ in the binomial expansion of $\left(2x^3+\dfrac3x\right)^{10}$ is $5^{10}-\beta\cdot3^9$, then $\beta$ is equal to\_\_\_\_

Answer: 83

Accepted Answer

$$T_{r+1}=\binom{10}{r}2^{10-r}3^r\,x^{30-4r}$$

Summing all $r=0$ to $10$ gives $(2+3)^{10}=5^{10}$; subtracting the $r=8,9,10$ terms (non-positive powers):

$$5^{10}-\left[\binom{10}{8}2^23^8+\binom{10}{9}2\cdot3^9+\binom{10}{10}3^{10}\right]=5^{10}-\left[60\times3^9+20\times3^9+3\times3^9\right]=5^{10}-83\times3^9$$

So $\beta=83$.

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