Use generating function to solve question

Question

Use generating function to determine the number of different ways 10 identical balloons can be given to 4 children if each children receives at least 2 balloons.

Answer 1

All balloons are identical, and we're distributing it among $4$ children, it is clearly a case of combination with repetition.

The given constraint is that each of the child receives at least $2$ balloons. Mathematically we can write:

$c_1 + c_2 + c_3 + c_4 = 10$, where $c_i \geq 2$

($c_i$ is the number of balloons the $i^{th}$ child got)

From, $c_i \geq 2 \implies c_i - 2\geq 0$

Let, $c_i - 2 = b_i \geq 0$

Now, we can rewrite the equation $c_1 + c_2 + c_3 + c_4 = 10$ as follows

$b_1 + b_2 + b_3 + b_4 = 2$, where $b_i \geq 0$

So, the answer to the question is the number of solutions possible for the above equation, which is

$${ 4 + 2 - 1 \choose 2 } = {5 \choose 2} = 10$$

Comments

Nice explanation.

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What is $\mathbf{b_i}$ here?

Answer 2

There are 4 children and 10 balloon so sum of total distribution will 10 always.

x1 + x2 + x3 + x4 = 10 (where x1,x2,x3,x4 ≥ 2)
every children should get 2 balloon so subtract the 8 balloon from total now only two ball balloon remaining which is to be distributed among 4 children.
x1 + x2 + x3 + x4 = 2 

             => ^n-1+rC_r   = ^4-1+2C₂ = ⁵C_{2 =} 10 different ways it can be distributed.