Combinatorics: solving for coefficient of x^n term

In summary, the conversation is about a person struggling to find the coefficient of the x^10 term in the equation f(x)=(x+x^2+x^3+x^4+x^5+x^6)^3. They mention a previous problem where the answer was 27 and they need to use a similar method to verify their answer. They mention starting with solving x^3/((1-x)^3) and using the binomial theorem. The person then receives a hint and realizes they were doing the binomial theorem wrong. This helps them solve the problem.
  • #1
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Hi, I'm currently taking a Discrete Mathematics class and cannot seem to work out this one problem, we need to find the x^10 term in order to determine its coefficient of the equation f(x)=(x+x^2+x^3+x^4+x^5+x^6)^3 I know the answer is to be 27 from a previous problem (we are to use this method to verify our answer) but I can't seem to figure it out. I started with thinking of trying to solve x^3/((1-x)^3)-the sum of x^n starting with n≥7, but that doesn't seem to be working. Any suggestions would be appreciated!
 
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  • #2
Hint:

[tex]f(x) = x^3 \left( \frac{1-x^6}{1-x} \right) ^ 3[/tex]
 
  • #3
Ahhh...I was confused for a second but I was doing the binomial theorem wrong. Thank you so much that helped tremendously!
 

What is combinatorics?

Combinatorics is a branch of mathematics that deals with the study of counting and arrangements of finite discrete objects. It involves the analysis of various techniques and formulas for solving problems involving combinations and permutations.

What is the coefficient of the x^n term?

The coefficient of the x^n term is the numerical value that multiplies the variable x raised to the nth power in a polynomial expression. It represents the number of combinations or arrangements of n objects.

How do you solve for the coefficient of the x^n term in a polynomial expression?

To solve for the coefficient of the x^n term, we can use the formula nCr = n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects being selected. This formula applies to combination problems, while permutations can be solved using the formula nPr = n! / (n-r)!.

What is the difference between combinations and permutations in combinatorics?

Combinations and permutations both involve arranging objects, but they differ in the order in which the objects are arranged. In combinations, the order does not matter, while in permutations, the order does matter. For example, the combination of choosing 3 letters from the word CAT would be the same regardless of the order (ACT, CTA, TAC), while the permutations would be different (ACT, CAT, TAC, TCA, CTA, ATC).

How is combinatorics used in real life situations?

Combinatorics has many real-life applications, such as in probability and statistics, computer science, genetics, and cryptography. It can be used to calculate the number of possible outcomes in a game of chance, to analyze the efficiency of algorithms, to study genetic traits and their combinations, and to create secure encryption methods for data protection.

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