I must "use the binomial theorem to determine the coefficient x^14 in the expansion of (3x^2-1/3)^16".

I find that:
n=16
r=14
a=3x^2 (I believe)
b=-1/3

I write that (sorry but the matrix brackets I cannot make):

(16) (3x^2) ^(16-14+1) (-1/3)^13
(13)

= (16) (3x^2)^3 (-1/3)^13
(13)

= (16!/(16-13)!13!) (1/? x^6)

This part is how I am getting confused.

Can somebody help me please to continue to solving this question? I must "expressing the coefficient as a fraction in lowest terms" but I don't think I have calculated well above to make the fraction to continue. The book is not showing this kind of difficult examples before we must do the assignment!

I must "use the binomial theorem to determine the coefficient x^14 in the expansion of (3x^2-1/3)^16".

I find that:
n=16
r=14
a=3x^2 (I believe)
b=-1/3

I write that (sorry but the matrix brackets I cannot make):

(16) (3x^2) ^(16-14+1) (-1/3)^13
(13)

= (16) (3x^2)^3 (-1/3)^13
(13)

= (16!/(16-13)!13!) (1/? x^6)

This part is how I am getting confused.

Can somebody help me please to continue to solving this question? I must "expressing the coefficient as a fraction in lowest terms" but I don't think I have calculated well above to make the fraction to continue. The book is not showing this kind of difficult examples before we must do the assignment!

In your case \(\displaystyle x = 3x^2\) and \(\displaystyle y= -\frac{1}{3}\)

Since \(\displaystyle -\frac{1}{3}\) does not contain an x term then raising it to the power so the power is unaffected.

As we have an \(\displaystyle x^2\) term the coefficient of x^14 will be given when n=7. Use the binomial theorem for the 7th term to find the coefficient

From the binomial theorem \(\displaystyle {16 \choose 7}(x^2)^7 \cdot \left(-\frac{1}{3}\right)^{16-7}\)

Hence the coefficient is \(\displaystyle {16 \choose 7} \cdot \left(-\frac{1}{3}\right)^{9}\).

Be aware that \(\displaystyle {16 \choose 7} = \frac{16!}{9!7!}\) whereas \(\displaystyle \left(-\frac{1}{3}\right)^{9}\) is simply minus one-third raised to the 9th power

I must "use the binomial theorem to determine the coefficient x^14 in the expansion of (3x^2-1/3)^16".

I find that:
n=16
r=14
a=3x^2 (I believe)
b=-1/3

I write that (sorry but the matrix brackets I cannot make):

(16) (3x^2) ^(16-14+1) (-1/3)^13
(13)

= (16) (3x^2)^3 (-1/3)^13
(13)

= (16!/(16-13)!13!) (1/? x^6)

This part is how I am getting confused.

Can somebody help me please to continue to solving this question? I must "expressing the coefficient as a fraction in lowest terms" but I don't think I have calculated well above to make the fraction to continue. The book is not showing this kind of difficult examples before we must do the assignment!