Kenneth Rosen Edition 6th Exercise 6.4 Question 13 (Page No. 440)

Question

Use Generating function to determine,the number of different ways $10$ identical balloons can be given to four children if each child receives atleast $2$ ballons?

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Ans given 

$(x^{2}+x^{3}+.........................)^{4}$

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But as there is a upper bound 

I think answer will be

$(x^{2}+x^{3}+.........................+{\color{Red} {x^{10}}})^{4}$

Which one is correct? plz confirm

Comments

Sorry,i'm not able to make $x^{2}?$

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Now i got the mistake which you made in your solution.

Here $k1+k2+k3=4$ you took 3 variables while the variables are four.

$k1+k2+k3+k4=2$  now it will be (4+2-1)C2 = 5C2 =10  and answer is 10

this entire solution given by you is wrong:

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Now i edit thanks
and answer is $10$

$[x^{2}]:(1+x+x^2)^4$

$[x^{2}]:\left\{\frac{1-x^3}{1-x}\right\}^4$

$[x^2]:(1-x^3)^4.(1-x)^{-4}$

$(a+b)^{n}=\sum_{k=0}^{n}\binom{n}{k} a^{n-k}.b^{k}$

$(a+b)^{-n}=\sum_{k=0}^{\infty}\binom{-n}{k} a^{-n-k}.b^{k}=\sum_{k=0}^{\infty}(-1)^k\binom{n+k-1}{k} a^{-n-k}.b^{k}$

$[x^2]:\sum_{k=0}^{4}(-x^3)^k.\sum_{k=0}^{\infty}(-1)^k.\binom{3+r}{r}(-x)^r$

find the coefficient of $x^2:$ Put $k=0,r=2$

                                 $[x^2]:\binom{3+2}{2}=\binom{5}{2}=10$
thanks

Answer 1

$Ans: 10$

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$[x^{10}]\ in\ (x^2+x^3+x^4+.....)^4$

$[x^{2}]\ in\ (1+x+x^2+x^3.....)^4$

$[x^{2}]\ in\ \dfrac{1}{(1-x)^4}$

$[x^{2}]\ in\ \sum_{k=0}^{\infty }\ ^{3+r}C_r(x^r)$

$Coeff\ of\ x^2=\ ^5C_2=10$