Tagged Questions

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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For counting permutations with identical objects, why does dividing nPr by the factorial of the number of identical objects give the correct answer?

I can find plenty of sites that say that this works, but I can't seem to find an explanation for why it works. I'm rather stumped.
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Expected sum value of permutaion

We have a set(A) of N elements. Let's assume elements are e1,e2,e3..etc. Value of each element can be 0 or 1. Another set of N elements(set B) are given, ...
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Linear Algebra - Permutations

Is it possible to multiply two permutations of different lengths together? If so how would you go about doing it?
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Suppose a coin in tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences…

Suppose a coin is tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences are there in which there are at least $5$ tails in a row? I know this is Permutation with repetition. My ...
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How many ways to arrange these gifts? (Inclusion-exclusion\derangement)

Each one of 30 people has bought 2 identical presents for the poor (every person's gifts are different from everyone else's). All the gifts were put in a large bag. In turns, 30 poor people ...
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There are $7$ empty seats on a bus and four people get on. How many different ways can they be seated?

There are $7$ empty seats on a bus and four people get on. How many different ways can they be seated? Would it be ${}^7C_{4}$ or ${}^7P_{4}$?
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Using the Binomial Identity, prove that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Using the Binomial Identity, prove that: $${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$$Because this is in the form of a Binomial Coefficient, I can break down the LHS ...
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How many different ways can you choose a group of $4$ people?

You have a total of $9$ people to choose from. Of these $9$ people you are supposed to create a group of $4$. How many different ways can the new group look? This is my reasoning: To the new group, ...
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How does $9\choose 4,3,2$ $=8$ $7\choose 4$

Can someone please explain to me how $9\choose 4,3,2$$=8$$7\choose 4$? From my understanding $9\choose 4,3,2$$=$$9\choose 4$$5\choose 3$$2\choose 2$$=$$9\choose 4$$5\choose 3$$\cdot 1$ But for ...
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Counting permutations in $S_n$ with $1,2,..,k$ all in same cycle

The number of permutations in $S_n$ for which the first $k$ items $1,2,...,k$ are all in the same cycle can be shown (by a somewhat tedious argument) to be $n!/k.$ I'm looking for less computational ...
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Partition of natural number not equal to factorial

I wish to prove the following statement so I can use it as a lemma for a group theory result. To be honest I have not tried much yet, my intuition tells me this is going to be connected to the ...
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Determining Permutation/Combinations with Bit Strings

I've got a discrete math problem on my hands...I'm trying to understand the steps behind working with bit strings; specifically, how a bit string of x length has "at least" or "exactly" a certain ...
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permutations of n objects

Does the number of permutations of $n$ objects, $r$ alike of one kind and $n−r$ alike of another kind, always equal the combinations of n different objects taken $r$ at a time? Explain. I know ...
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Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given ...
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Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
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Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
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how many 5 digit numbers are there with distinct digits?

I found this question on gre forum, it's answer was given by this expression: $9\cdot9\cdot8\cdot7\cdot6$ which I heard in school as well. What I tried to do was: for numbers from index $4$ to ...
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Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...
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How many straight lines can be made between 10 points such that 4 of them are colinear?

So i know how to get the answer. We just have to find $C(10,2)$ and subtract $C(4,2)$ and add 1. We are basically counting all the points between co-linear points as 1. So the question is why we are ...
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How many distinct ways can the number be written as product of $3$ factors?

How many distinct ways can the number $126$ be written as a product of $3$ positive integer factors? I found that the prime factors are $126=2\times3\times3\times7$. But how to get number of ...