Find the coefficient "a" of the given term in the expansion of the binomial.

Question

Binomial: (2x - 5y)¹⁰ Term: ax⁵y⁵

^{I'm supposed to come up with a large number but I have no clue how to do that}

Answer 1

I answered another of your questions earlier with a similar response. I was hoping you would use it to figure out how this works:

We can typically expect a "10th power" polynomial to have 11 terms. Each term includes a contribution from the first term in the binomial, in this case "2x" (let's call the coefficient contribution "a"), the last term in the binomial, in this case "-5y" (let's call the coefficient contribution "b"), and from Pascal's triangle (let's call that contribution "p"). A 10th degree polynomial (with first binomial variable "x" and last variable "y") would generically look like this:

p₁a₁₀x¹⁰b₀y⁰ + p₂a₉x⁹b₁y¹ + p₃a₈x⁸b₂y² + p₄a₇x⁷b₃y³ + p₅a₆x⁶b₄y⁴ + p₆a₅x⁵b₅y⁵ + p₇a₄x⁴b₆y⁶ + p₈a₃x³b₇y⁷ + p₉a₂x²b₈y⁸ + p₁₀a₁x¹b₉y⁹ + p₁₁a₀x⁰b₁₀y¹⁰

So the term we are looking for is what I have listed as p₆a₅x⁵b₅y⁵ so we just need p₆, a₅, and b₅. The coefficient we are seeking will just be the product of those three.

In this case, the coefficient of the first term in the binomial is "2" so a₅ will be 2⁵ because the "x" variable is raised to the 5th power. So a₅ = 32

In this case, the coefficient of the last term in the binomial is "-5" so b₅ will be (-5)⁵ because the "y" variable is raised to the 5th power. So b₅ = -3125.

Using the 10th row on Pascal's triangle, we get:

       1        10      45      120    210    252    210    120    45    10    1

We can see that p₆ = 252.

So the coefficient we seek is (32)(-3125)(252) = -25200000