Find the coefficients of X^12y^6 in the expansion of (x^3-3y)^10

Find the constant term in the expansion of (y-1/2y)^10

(x^{3}-3y)^{10}

Here is one way to go about it by listing out the terms.

The first term will be x^{30}

The coefficient of the second term will be the exponent from the first term.

30x^{27}y

Reduce the exponents of x by 3 and add 1 to the exponent of y.

Now, To find the next coefficient, multiply the coefficient by the exponent and divide by 1 more than the exponent of y. 30*27/2=405

So, the next term is 405x^{24}y^{2}

The next would be 405*24/3=3240

3240x^{21}y^{3}

Continue in this manner until you get to y^6. Not too much further.

So far, we have x^{30}-30x^{27}y+405x^{24}y^{2}-3240x^{21}y^{3}+....

Remember to alternate your signs.

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For the second one, is that (y-\frac{1}{2}y)^{10}?

If so, look at it. It is just (\frac{y}{2})^{10}. Now, can you see the constant term?