# Pascal's Triangle and Binomial Theorem

1. Oct 2, 2007

### SurferStrobe

1. Evaluate the numbers for the coefficient of x4y9 in the expansion of (3x + y)13.

2. The Binomial Theorem states that for every positive integer n,
(x + y)n = C(n,0)xn + C(n,1)xn-1y + ... + C(n,n-1)xyn-1 + C(n,n)yn.

3. I understand that the coefficients can be found from the n row of Pascal's triangle, where n = 13. Using the binomial theorem, my approach (which I'm not sure about) is:

The coefficient is 3 * C(13,9) = 3 * 715 = 2145.

Am I going about this correctly? Sorry if I didn't expand on the proof.

surferstrobe

2. Oct 2, 2007

### EnumaElish

This is the 5th term from the end (if you start from x^13 then diminish x's power), so is C(13,9) consistent with the formula?

And, you should have 34.

Last edited: Oct 2, 2007
3. Oct 2, 2007

### SurferStrobe

I was just going to say, I forgot the exponent of x, so should be 34 * C(13,9).

I used the exponent of y for the combination term, because I knew that

C(13,9) = C(13,13-9) = C(13,4).

Therefore,

34 * C(13,9) = 81 * 715 = 57,915.

Is this better?

4. Oct 3, 2007

### EnumaElish

In the formula x^k is associated with C(n,k-1) isn't it?

5. Oct 3, 2007

### SurferStrobe

EnumaElish,

Thank you for helping me to better understand the relationships between the coefficients, exponents, and expressions in this area of discrete mathemetics!

surferstrobe

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