What is the sum of the coefficients of the terms containing $x^2t^3$ in the given expansion. 
It is given that $(x^2 +y +2t +3k)^{10}$. What is the sum of the coefficients of the terms containing $x^2t^3$  in the given expansion. For example the terms are $x^2y^4t^3k^2$ etc.

I said that let $(x^2 +2t)$ be $"a"$ and $(y +3k)$ be $"b"$. Then, $(x^2 +y +2t +3k)^{10}=(a+b)^{10}$.
If we are looking for  $x^2t^3$, then the exponential of $a$ must be $4$. Then, the exponential of $b$ is $6$.
Then, the coefficient of $x^2t^3$ is $4 \times 2^3=32$. Now, we have $(y +3k)^6$ as $b$. If we find the sum of the all coefficients in $(y +3k)^6$, we can handle the question.
The sum of the all coefficients in $(y +3k)^6$  is $4^6$.
Then, answer is $32 \times 4^6$
Is my solution correct ?

Here's a hint to simplify the calculation. Suppose you were to fully expand $(x^2+y+2t+3k)^{10}$ and then drop all but those terms containing $x^2 t$. What results is an expression of the form
$$ax^2 t+bk x^2 t+cy x^2 t+dk^2 x^2 t+\cdots=(a+bk+cy+dk^2+\cdots)x^2 t$$
Summing up these coefficients yields $a+b+c +d+\cdots$.
What's the relation between this sum and the polynomial in $y,k$ written above? (This reduces the problem to finding the coefficient of $x^2 t$ in some polynomial in $x,t$.)
