Use generating functions to determine the number of different ways 15 identical st\uffed animals

step 1 We are asked to use generating functions to solve a problem. We are asked to find the number of

Use generating functions to determine the number of...

Key Concepts

Generating Functions

A generating function is a formal power series where the coefficients correspond to the number of ways to achieve certain outcomes. It provides a compact representation of sequences and is used to encode combinatorial problems into algebraic form for systematic analysis and solution extraction.

Restricted Integer Partitions

Restricted integer partitions involve dividing a whole number into a sum of integers that meet specific constraints, such as minimum or maximum values for each part. This concept is fundamental in problems where items must be distributed according to given upper and lower bounds.

Coefficient Extraction

Coefficient extraction is the process of determining a specific term’s coefficient in the expansion of a generating function. This coefficient represents the number of valid combinations or ways to distribute objects under the imposed restrictions.

Combinatorial Analysis

Combinatorial analysis is the study of counting, arranging, and structuring elements within a set according to particular rules or conditions. It relies on tools like generating functions to systematically solve problems involving distributions and partitions with constraints.

Transcript

00:01 We are asked to use generating functions to solve a problem.

00:07 We are asked to find the number of different ways 15 identical stuffed animals can be given to six children so that each child receives at least one, but no more than three stuffed animals.

00:23 So for each child, the function is going to be x plus x squared plus x cubed.

00:38 Since the child can have at least one stuffed animal, but no more than three stuffed animals.

00:49 And since we have six children, our generating function is going to be x plus x squared plus x cubed to the sixth power.

01:11 And this can be written as x to the sixth times one plus x plus x squared to the sixth power, which can further be simplified to x to the sixth times, and this is 1 minus x cubed over 1 minus x to the 6th power.

01:52 And this can be written as x to the 6th times 1 minus x cubed to the 6th over or times 1 minus x to the negative 6th power.

02:15 And using the extended binomial theorem, this is equal to x to the sixth times the sum from k equals 0 to infinity of 6 choose k negative x cubed to the kth power times the sum from k equal 0 to infinity of 6 plus k, k, minus 1 choose k, x to the kth power.

03:25 This is equal to x to the 6th times the sum from k equals 0 to infinity of 6 choose k, and negative 1 to the kth power times x to the 3k, times the sum from k equal 0 to infinity of 5 plus k, choose k x to the kth power we want to find the coefficient of x to the 15th since there are 15 stuffed animals and so if we take this k to be m and this k to be n we have that 6 plus 3m plus n must be equal to 15 which can be written as 3m 3m 3m plus n must equal 9.

05:02 And so to find these values, it's plug in different values of m.

05:10 So if m is equal to 0, we have that n has to equal 9.

05:17 If m is equal to 1, we have that n has to equal 6.

05:26 If m is equal to 2, then n is equal to 3.

05:35 And if m is equal to three, then n has to equal zero.

05:55 And so we have the coefficient of x to the 15th is going to be a sum of these coefficients, which are if m is zero and n is nine, this is six choose zero times negative one to zero of power times five plus nine is four times, choose 9 plus if m is 3 or m is 1 and is 6 we have 6 choose 1 negative 1 to the first and n is 6 so we have 11 choose 2 negative 1 to the 2 and n is 6 so we have 11 choose 6 plus m is 2 we have 6 choose 2 negative 1 to the 2 and n is 3 so we have 3 plus 5 is 8, choose 3 plus, and then if m is 3, n is 0, and we have the coefficient is 6, choose 3, negative 1 to the third power, and 5 choose 0, this is equal to 6 choose 0 is 1, negative 1 to the 0 is 1.

08:09 And so we have 14 choose 9, which is equal to 14 factorial over 9 factorial times 5 factorial, plus 6 choose 1 is 6 times negative 1, so this is a minus 6, and then 11 choose 6 is 11 factorial over 6 factorial times 5 factorial, plus 6 choose 2, this is 6 factorial over 2 factorial, this is 6 factorial, times 4 factorial and negative 1 squared is 1 and 8 choose 3 is 8 factorial over 3 times 5 factorial plus this is going to be a minus since negative 1 cubed is 0 is 1 minus minus 6 factorial over 3 factorial squared times and 5 choose 0 is simply 1...

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