# Binomial expansion question that I cannot fathom

1. Dec 6, 2012

### Originaltitle

1. The problem statement, all variables and given/known data

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???

2. Relevant equations

Binomial expansion formula.

3. The attempt at a solution

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

2. Dec 6, 2012

### Ray Vickson

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.

3. Dec 6, 2012

### 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.

4. Dec 6, 2012

### Ray Vickson

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

5. Dec 6, 2012

### 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.

6. Dec 7, 2012

### Curious3141

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.

7. Dec 8, 2012

Thanks.