Binomial expansion question that I cannot fathom

[Originaltitle]

Homework Statement

It says: Determine the coefficient of p⁴q⁷ in the expansion of (2p-q)(p+q)¹⁰.

I can find the coefficient of p⁴q⁶ in the expansion of (p+q)¹⁰ but how am I to find it for (2p-q)(p+q)¹⁰?

Homework Equations

Binomial expansion formula.

The Attempt at a Solution

[Coefficient of p⁴q⁶ in the expansion of (p+q)¹⁰ = (¹⁰C₆) x (p)⁴ x (q)⁶.]

 

[Ray Vickson]

Homework Statement

It says: Determine the coefficient of p⁴q⁷ in the expansion of (2p-q)(p+q)¹⁰.

I can find the coefficient of p⁴q⁶ in the expansion of (p+q)¹⁰ but how am I to find it for (2p-q)(p+q)¹⁰?

Homework Equations

Binomial expansion formula.

The Attempt at a Solution

[Coefficient of p⁴q⁶ in the expansion of (p+q)¹⁰ = (¹⁰C₆) x (p)⁴ x (q)⁶.]

Write
$$(2p-q)(p+q)^{10} = 2 p (p+q)^{10} - q (p+q)^{10},$$ then find the coefficients of p^4 q^6 in each term separately.

 

[Originaltitle]

But they're asking for the coefficient of p^4q^7, not p^4q^6. BUT 4 + 7 = 11 and 11 is not the power on the original bracket. The powers on p and q must add up to 11, but they can't over here.

 

[Ray Vickson]

But they're asking for the coefficient of p^4q^7, not p^4q^6. BUT 4 + 7 = 11 and 11 is not the power on the original bracket. The powers on p and q must add up to 11, but they can't over here.

I have told you how I would do the problem if I had to.

 

[Originaltitle]

You did but they're not asking for what you're doing. They're asking for the coeff. of p^4q^7, not p^4q^6 which is what you're finding.

 

[Curious3141]

You did but they're not asking for what you're doing. They're asking for the coeff. of p^4q^7, not p^4q^6 which is what you're finding.

Use Ray Vickson's method. For the binomial $(p+q)^{10}$, find the coefficient of $p^3q^7$ for the first product, and $p^4q^6$ for the second product.

What happens when you multiply the first product by $2p$, and the second by $q$? Now do the subtraction.

 

[Originaltitle]

Thanks.