Pascal's Triangle and Binomial Theorem

[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)x^n-1y + ... + C(n,n-1)xy^n-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

 

[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.

 

[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?

 

[EnumaElish]

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?

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

 

[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