Generating Functions

[mahjk17]

I am having trouble finding the coefficient of this problem, "A basket of fruit has 20 pieces: apples, pears, oranges, and grapes.
How many different ways can we prepare the basket if there is at least one apple in the basket, the basket cannot have more than three pears, and the number of
oranges must be a multiple of four?" I already found the generating function to be x/(1-x)^3 but I cannot seem to find the coefficient. I keep getting 22 choose 3 which is obviously wrong. Can anyone help me find it ?? Thank you!!

 

[mmm4444bot]

the number of oranges must be a multiple of four

Zero is a multiple of four. Are you considering baskets that have no oranges?

 

[galactus]

Using the binomial series, note that:

\(\displaystyle \displaystyle\frac{x}{(1-x)^{3}}=\sum_{k=0}^{\infty}\binom{-3}{k}(-1)^{k}x^{k+1}\)

This can also be written as:

Note that \(\displaystyle \displaystyle \frac{x}{(1-x)^{n}}=x+\binom{1+n-1}{1}x^{2}+\binom{2+n-1}{2}x^{3}+\cdot\cdot\cdot + \binom{r+n-1}{r}x^{r+1}+\cdot\cdot\cdot\)

 

[mahjk17]

mmm444bot, I am not considering 0 as a chose..
And galactus you are right : ), thank you very much !!!! You are a genius! Thank you again!

 

[mahjk17]

One last question galactus, because we are looking for the total arrangement of objects we need to replace n with 20 from the open form you provided me because 20 is the total???? So our answer for the coefficient should be taken from x^3 which coefficient will equal to 21 choose 2? So our answer is 21 choose 2??? Or am i wrong?