Combinatorics: Exponential Generating Functions

[Shoney45]

Homework Statement

Find the number of ways to place 8 toys amongst 4 children where 1 child gets at least two toys.

Homework Equations

(x^2/2! + x^3/3! + x^4/4! +...) = e^x-1-x

(1 + x + x^2/2! + x^3/3! +...)³ = e^3x

The Attempt at a Solution

[(x^2/2! + x^3/3! + x^4/4! +...) = e^x-1-x] represents the child who gets at least two toys.

[(1 + x + x^2/2! + x^3/3! +...)³ = e^3x] represents the other three children.

Thus, [e^x-1-x] * [e^3x] = e^x-xe^3x
= (summation)1*x^r/r! - x*(summation)3^r*x^r/r!.

So the coefficient of x⁸/8! = 1 - (x*3⁸).

What I'm unsure of is the 'x' in the final coefficient since that makes my answer a variable coefficient, and not an exact answer to the question "How many ways are there..."

 

[mighty2000]

Thank you for posing this interesting problem. I would approach this problem by first understanding the fundamental principles of combinatorics and then applying them to this specific scenario.

Firstly, we must recognize that the number of ways to place 8 toys among 4 children can be represented by the combination formula, nCr = n!/(r!(n-r)!). In this case, n = 8 and r = 4, so the number of ways to place the 8 toys among the 4 children is 8!/(4!(8-4)!) = 8!/(4!4!) = 70.

However, the problem also specifies that one child must receive at least two toys. This means we must subtract the number of ways where all 4 children receive only 1 toy each. This can be represented by (1*1*1*1) = 1, since each child can only receive 1 toy. Therefore, the final answer would be 70 - 1 = 69 ways.

Alternatively, we can use the principle of inclusion-exclusion to solve this problem. This principle states that the total number of ways to place n objects among k groups, where at least one group receives at least m objects, is given by:

nCr - (k-1)C1 * (n-m)Cr + (k-1)C2 * (n-2m)Cr - (k-1)C3 * (n-3m)Cr + ...

Applying this to our problem, we have n = 8, k = 4, and m = 2. Thus, the total number of ways is:

8C4 - 3C1 * 6C4 + 3C2 * 4C4 - 3C3 * 2C4 = 70 - 3*15 + 3*1 - 1 = 69 ways.

I hope this helps you understand the problem better and gives you some ideas on how to approach similar problems in the future. Remember, as a scientist, it is important to have a strong understanding of fundamental principles and how to apply them to solve problems. Keep up the good work!