Solving Generating Function when there is condition on two variables.

Question

" Find the number of ways of giving 10 identical gift boxes to 6 people : A, B, C, D, E, F in such a way that total number of boxes given to A and B together does not exceed 4. "

I tried it in this way :

[$x^{10}$] $(1+x^{1}+...+x^{4})^{2}*( 1+x^{2}+....)^{4}$

But I am not sure if its right or not , can anyone help me with the condition on A and B i.e say $x_1+x_2\leq 4$

Ankit Kumar · Accepted Answer · 2018-12-22T13:18:51.460

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What you've to find is $x_1+x_2+x_3+x_4+x_5+x_6=10$ (I've replaced A-F by 1-6). Further, $x_i\geq 0$ and $x_1+x_2\leq 4$.

Required answer= $$\sum_{i=0}^{4}(Coeff.\ of\ x^i\ in \ (x^0+x^1+\cdots x^4)\cdot (x^0+x^1+\cdots x^4))\cdot(Coeff.\ of \ x^{10-i}\ in \ (x^0+x^1+\cdots x^{10})^4)$$ Can you solve it now?

In open form, it can also be written as: $$(Coeff.\ of\ x^0\ in \ (x^0+x^1+\cdots x^4)^2)\cdot(Coeff.\ of \ x^{10}\ in \ (x^0+x^1+\cdots x^{10})^4)$$$$+(Coeff.\ of\ x^1\ in \ (x^0+x^1+\cdots x^4)^2)\cdot(Coeff.\ of \ x^{9}\ in \ (x^0+x^1+\cdots x^{10})^4)$$$$+(Coeff.\ of\ x^2\ in \ (x^0+x^1+\cdots x^4)^2)\cdot(Coeff.\ of \ x^{8}\ in \ (x^0+x^1+\cdots x^{10})^4)$$$$+(Coeff.\ of\ x^3\ in \ (x^0+x^1+\cdots x^4)^2)\cdot(Coeff.\ of \ x^{7}\ in \ (x^0+x^1+\cdots x^{10})^4)$$$$+(Coeff.\ of\ x^4\ in \ (x^0+x^1+\cdots x^4)^2)\cdot(Coeff.\ of \ x^{6}\ in \ (x^0+x^1+\cdots x^{10})^4)$$

edited Dec 22 '18 at 13:18

answered Dec 22 '18 at 12:58

Ankit Kumar

1,874

And yes, you can take the second term to infinity instead of just $x^{10}$, as you've written since higher powers won't affect your answer – Ankit Kumar Dec 22 '18 at 13:00
then whats wrong in my 1st part ? – CHETAN RAJPUT Dec 22 '18 at 13:06
and if its wrong, will you please give me the correct way of doing it using generating functions, i mean the way i wrote it.. thats familiar for me to solve. thanx – CHETAN RAJPUT Dec 22 '18 at 13:07
You've not taken $x_1+x_2\leq 4$. What you've taken is $x_1\leq 4$ and $x_2\leq 4$. Note that the first term in your expression can give you $x^8$, $x^4\cdot x^4$, which shouldn't be the case right? – Ankit Kumar Dec 22 '18 at 13:09
I've also used generating functions man! I'm also finding coefficient of $x^10$, just like you. It's just that my solution has 5 terms, while ours had just one! Further, I don't even think you can solve it using just one term – Ankit Kumar Dec 22 '18 at 13:12
Yeah i had that Doubt, so will you update your answer the way i wrote it. In form of generating function only, ( dont involve summation ) – CHETAN RAJPUT Dec 22 '18 at 13:14
Ohhh... got it ..., but in that case also , see for your 1st term it is 1( x^0+..+x^4)(x^0+..+x^4) then wont it generate x^8 ? – CHETAN RAJPUT Dec 22 '18 at 13:17
I don't care about the coefficient of $x^8$. See again, I've taken the coeff. of $x^i$, and $0\leq i \leq 4$ – Ankit Kumar Dec 22 '18 at 13:20
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Okay. I got that. I haven't came across this case so I am not fully getting it. I ll cross check the answer then analyse it. Thanx – CHETAN RAJPUT Dec 22 '18 at 13:25
if you have any reference to this please share – CHETAN RAJPUT Dec 22 '18 at 13:27
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https://issuu.com/imsf/docs/mcs-033/25 It has unit 2, generating functions – Ankit Kumar Dec 22 '18 at 13:37
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Thank you for sharing it. I got 2121 , check if its fine. – CHETAN RAJPUT Dec 22 '18 at 15:03
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Thanx. answer is correct. – CHETAN RAJPUT Dec 22 '18 at 16:39
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@CHETANRAJPUT you're welcome – Ankit Kumar Dec 22 '18 at 16:52

Solving Generating Function when there is condition on two variables.

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