# Questions tagged [combinations]

Combinations are subsets of a given size of a given finite set. All questions for this tag have to directly involve combinations; if instead the question is about binomial coefficients, use that tag.

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### Permutations of n random variables using 3 alphabets [duplicate]

I am trying to work out the following: The number of possible arrangements of (S1, S2, ... Sn), where each Si can take one of 3 values (A, B, C), and where adjacent Si's do not have the same value. ...
41 views

### A committee of 11 members is to be formed from 8 males and 5 females with at least 6 males in the committee [duplicate]

Q) A committee of 11 members is to be formed from 8 males and 5 females. The number of ways the committee is formed with at least 6 males is? The general solution I see everywhere is to take 3 cases ...
45 views

### Probability of 2 individual people being put in the same group of 3 with 39 total people

I am trying to understand the possible number of combinations for two specific people (let's call them Person A and Person B) to be put in the same group of 3 if there are 39 total people to sort into ...
58 views

### I do not understand why do we divide with the k! that is used in nCk when multiplying two combinations or more combinations that has the same k value

I recently started studying Permutations and Combinations and following is one interesting question I came across. There are 8 students in a class. The class ...
35 views

### Counting Permutations with Product Rule

I am working on my homework and I solved this problem and I just wanted someone to double check to see if my answer makes sense because I don't trust chat gpt. 5.4.3: Lining up club members for a ...
49 views

### a group of 20 people, 12 men and 8 women, must elect a board of 4 people. On how many ways can this happen if the board is to have at least 1 woman?

a group of 20 people, 12 men and 8 women, must elect a board of 4 people. On how many ways can this happen if the board is to have at least 1 woman? I know that the amount of combinations as a whole ...
97 views
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### intuitively Understanding meaning of a Combinatorics problem to reach solution

I have been recently taking course on Combinatorics and landed on following problem, Here is the formal statement of problem: A room contains a single bulb and $(2^{2^{10}}+2^{2^9})$ identical ...
1 vote
44 views

### The interpretation of 'at least one' event in probability

In the following question, I don't understand why combination such as {20,19,18} is not even considered? Or it's already implied in some of the calculation below implicitly? (but it's not clear to me, ...
1 vote
56 views

### Why is the count of unique results from rolling two dice once 7C2?

By "unique results" I mean that the order of the two resulting numbers does not matter - a 1 and a 2 is the same as a 2 and a 1. I have used a spreadsheet to brute force the answer of 21, ...
25 views

### Discrete math example clarification (stars and bars)

Example 6.3.4 from Discrete Mathematics by Richard Johnsonbaugh 8th ed, page 285 (abridged): "Consider three books: a computer science book, a physics book, and a history book. Suppose that the ...
35 views

### How many matches can be made

Two schools should play one against the other table tennis. Five students represent each of these schools. Doubles should only play. Each pair from one school should play against each pair from the ...
66 views

### The Collector’s Problem

I have the following Question in my text book , where the Answer is given. I have got my own alternative Answer. I want to know whether that alternative Answer is Correct & Equivalent to the text ...
1 vote
33 views

### Probability of exactly $m$ bins being occupied

N indistinguishable balls have to be assigned to K bins (where each bin can have at most C balls). What is the probability that exactly m bins are occupied?
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### There are eighty items grouped into twenty. Each group contains 4 items. In how many ways can I combine 10 groups and the code to list them [closed]

In how many ways can 4items in each group be combined with 4 items of 9 other groups in the total of the eighty items
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### The table top is in the shape of a regular heptagon and diagonals are drawn on it. In how many ways can the figure be colored with two colors?

The table top is in the shape of a regular heptagon and diagonals are drawn on it as shown below: In how many geometrically different ways can the resulting figure be colored with two colors at our ...
31 views

### Ways to pick $k$ objects from $n$ objects of $p$ different kinds, where some are alike? [duplicate]

How many ways are there (in both case where order matters (permutations) and order does not matter (combinations) ) if one pick $k$ candidates from $n$ elements of $p$ different kinds where: $n_1$ are ...
1 vote
101 views

### In how many ways can the passengers be conveyed to the neighboring village from the train station?

25 passengers arrive at a railway station and proceed to the neighboring village. At the station there are 2 coaches accommodating 4 each & 3 carts accommodating 3 each. Find the number of ways in ...
59 views

### Find the number of nonnegative integer solutions combination and deck of cards question [closed]

I have a question about combinations from my homework. Question 1. Find the number of nonnegative integer solutions to $$(x_1+x_2+x_3)(x_4+x_5+x_6) = 20$$ How do I tackle this question? From my ...
1 vote
54 views

### Distributing $m\cdot n$ identical items into $n$ distinct boxes, with $m$ distinct positions in box

There are $n$ boxes and $m\cdot n$ identical items. Each box has $m$ positions where an item can be placed. The order matters here. It is a difference if an item is placed in position 1, 2, or 3 or if ...
53 views

### Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$

I want to find the number of positive integer solutions of the equations given by $$\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10.$$ I know the case that, for any pair of ...
118 views

### Unconventional Dice (PnC Question)

I am getting the answer as 48, but at some places, the answer given is 24. I am just fixing 1 at the top and 2 at the bottom. Rest 3,4,5,6 can be arranged in 3! ways and colours can be decided in ...
318 views

### In how many ways can four distinctive letters be posted in 6 post boxes such that any two go in same post box and remaining go to different boxes?

I have tried to solve it in two different ways. But I ended up with different answers. First method There are $\binom{4}{2}=6$ different $2$ letter combinations from a set of $4$ letters. With the ...
1 vote
64 views

### Probability that at least one bin has all balls of the same color

There are $32$ balls in total, equally split into red, green, blue, and white colors ($8$ balls each). I also have $4$ bins. What is the probability that in at least one of the bins, all the balls are ...
1 vote
43 views

### Subsets of three numbers from a set of n numbers such that no two subsets have a shared pair of numbers

I've had this problem for a long time. The problem goes as follows (it is a bit hard to say concisely): Let's say, for example, I have the numbers 1-9 (n=9) and am looking to arrange these numbers in ...
146 views

### Probability: Card distribution problem

A deck of 52 cards is equally dealt to 4 players. Find the number of ways to distribute the cards so that each player has exactly one card from each rank. [Note: A deck of 52 cards consists of 13 ...
42 views

### Combination problems with replacement

A restaurant has n items on its menu. During a particular day, k customers will arrive and each one will choose one item. The manager wants to count how many different collections of customer choices ...
49 views

### Solving Combinatorial Problem - Red Ball Distribution in Three Bins

I'm reaching out to the community for help in tackling a combinatorial problem that I'm currently stuck on. The problem revolves around distributing twelve identical red balls into three bins, labeled ...
219 views

### What is the total number of rectangles in a grid that do not include a circle?

I am having trouble solving this question where I have to find the total number of rectangles in the grid shown below that do not include any of the circles. I believe that this question can be done ...