Program segment question for combinatorics

In summary, the given segment of code uses nested loops to generate combinations of three elements from a set of n+2 elements. The value of counter is incremented by one for each combination generated. This can be used for study purposes, but is not an efficient method for generating combinations. It is equivalent to the binomial coefficient ${n+2 \choose 3}$.
  • #1
4
0
Consider the following segment where i,j,k,n and counter are integer variables and the value of n (a positive integer) is set prior to this segment.

counter := 0
for i := 1 to n do
for j :=1 1 to i do
for k :=1 to j do
counter := counter + 1

I am so lost as to what this program is even doing. Can anyone help explain what is does? I just need to know what it does upon first execution, second execution, etc.. and I may be able to solve it from there.
 
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  • #2
Magenta55 said:
Consider the following segment where i,j,k,n and counter are integer variables and the value of n (a positive integer) is set prior to this segment.

counter := 0
for i := 1 to n do
for j :=1 1 to i do
for k :=1 to j do
counter := counter + 1

I am so lost as to what this program is even doing. Can anyone help explain what is does? I just need to know what it does upon first execution, second execution, etc.. and I may be able to solve it from there.
It's a simple nested loop that will end up with counter = j*i*n, which is a waste of time since you could just do the multiplication. Where nested loops are useful is where you DO something useful inside the loops, so I assume this is just an example of a nested loop for study purposes.

EDIT: OOPS ... no, that's NOT what it is. I see now that the inner loops are using the outer loop variables for a limit, SO ... this is something like a bubble sort nested loop and can be used for something LIKE(*) # of combinations or permutations. I'd have to look more closely to see how it might relate to combinatorial forumlae, but basically it's just a nested loop which, as I said, uses the outer loop variable as the inner loop limit.

(*) LIKE is emphasized to mean "not necessarily exactly" since I have not looked at the results in detail.
 
Last edited:
  • #3
This gives you ##{n+2 \choose 3}##. You have n+2 elements going from 1 to n+2. first you pick the element 2+i from range ##[3;n+2]##, then element 1+j from range ##[2;1+i]##, then k from ##[1;j]##. You are choosing all such possible combinations and by making the next choice in the range smaller than the last choice, you guarantee that you don't pick the same triple many times in a different order. The shifting starting point is necessary so that you have something to pick next, if you pick the first element at the first step, there won't be any left to pick in the next step, because in your formula the next index goes up to the previous one. So you are getting how many different triples can you take from n+2, which is the binomial coefficient ##{n+2 \choose 3}##.
 

What is a program segment question for combinatorics?

A program segment question for combinatorics is a problem that requires the use of mathematical techniques to determine the number of possible outcomes or arrangements of a given set of elements. It involves applying principles of combinatorics, such as permutations and combinations, to solve the problem.

What are some common types of program segment questions for combinatorics?

Some common types of program segment questions for combinatorics include problems involving arrangements, selections, and distributions. For example, determining the number of ways to arrange a set of objects, the number of ways to select a subset of objects, or the number of ways to distribute objects into groups.

How do you approach solving a program segment question for combinatorics?

The first step in solving a program segment question for combinatorics is to carefully read and understand the problem, identifying the given elements and what is being asked. Then, depending on the type of problem, you can use techniques such as permutations, combinations, or the multiplication principle to determine the number of possible outcomes. Finally, make sure to double check your answer and consider any special cases or restrictions given in the problem.

What are some tips for tackling difficult program segment questions for combinatorics?

Some tips for tackling difficult program segment questions for combinatorics include breaking the problem down into smaller, more manageable parts, using diagrams or tables to organize information, and considering different approaches or techniques if you get stuck. It can also be helpful to practice solving similar problems and to seek help from a teacher or peer if needed.

How can understanding combinatorics be useful in other fields of science?

Combinatorics has applications in many fields of science, including computer science, statistics, and genetics. Understanding combinatorics can help in analyzing and organizing data, making predictions and calculations, and solving problems in these fields. It is also a fundamental concept in probability, which is essential in many scientific experiments and studies.

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