27 questions linked to/from How to use stars and bars?
876 views

Find the number of combinations such that sum of N numbers equals to M [duplicate]

Here is the problem: Let M (the sum) be 10 and N (the number of numbers) be 4. Some possible combinations are are (10, 0, 0, 0), (5, 3, 2, 0), (8, 1, 1, 0), (8, 2, 0, 0), (6, 2, 1, 1), etc. How do ...
174 views

How many ways so that $x+y+z\leq n$? [duplicate]

Can anyone help me out finding the number of solutions i.e. $(x,y,z)$ for the inequality $x+y+z\leq n$ where, $n$ is a constant positive value and $x,y,z$ are non-negative?
64 views

How many solutions are there to $x+y+z=n$? Need help understanding the answer. [duplicate]

I need some help. I do not understand how to get the answer (solution) to this question. I could not solve it, neither did it help when I saw the answer. This is a question from the chapter ...
51 views

Elementary Diophantine Equation in 5 variables [duplicate]

A student of mine has asked me to solve the following Diophantine equation, but it has been a long time since I looked at these problems. Here is the problem: Problem: Find the natural number ...
44 views

Positive integral solutions to any equation $x + y + z = n$ [duplicate]

I thought of using the balls and flags trick - where you use the flags to separate identical balls into different groups. But I am not able to discard the solutions in which I get $zeroes$. Please ...
37 views

Counting solutions to a five-variable equation [duplicate]

How many solutions (using only nonnegative integers) are there to the equation $$x_1+ x_2 + x_3 + x_4 + x_5 = 31\;?$$
15 views

Combination-Counting problem [duplicate]

How many different ways can 200 coins be distributed among 10 people so that: 1)everyone gets 0, or more, coins? 2)everyone gets 5, or more, coins? I know how to do question it they have an exact ...
2k views

In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits

In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits. My attempt:We should find number of two pairs that can be digits of this number for choosing the ...
2k views

Split a number into parts

In how many ways can a natural number $n$ be split into $m$ natural numbers (parts) where each part is less than $n$, the parts don't necessarily have to be equal, and all of them add up to $n$?
7k views

How do I calculate the number of unique permutations in a list with repeated elements? [duplicate]

I know that I can get the number of permutations of items in a list without repetition using (n!) How would I calculate the number of unique permutations when a ...
1k views

How to convert a problem to a stars and bars problem?

Continued question from here. With certain questions I have $x_i$ being constrained by various different inequalities, I want to know how to remove these from the problem, to bring me back to a ...
274 views

Math probability that “5-out-of-36” lottery draw has at least one pair of numbers with difference = 1.

Given a $5$ out of $36$ lottery ($5$ unique numbers out of pool of $36$ numbers ranging $[1,2,…,36]$). How to calculate probability that a draw has at least one pair of consecutive numbers (like ...
244 views

Combinatorics for a 3-d rotating automaton

Let's suppose that we have some kind of special 3-dimensional rotating automaton. The automaton is capable to generate rotation about selected $X$ or $Y$ or $Z$ axis (in a current frame) in steps by ...
Finding number of non negative integral solutions of $a+2b+3c+4d=20$
How to find out the number of non negative integral solutions of an equation containing 4 variables, for eg, say, ${a+2b+3c+4d=20}$? I mean, we can calculate it quite easily for equations containing ...
Question : I wanted to find out the number of ways all the 7 numbers below could be assigned positive integers which can sum up to 42.$$x_1+x_2+x_3+x_4+x_5+x_6+x_7=42$$ the catch in this problem is ...