-1
votes
0answers
21 views

Simplifying a sum with factorials and permutation counts

I am able to simplify it to a form of summation $$\frac{{n\choose r}}{K {a\choose b}}$$ but not able to proceed any further.
0
votes
1answer
26 views

Count no. Of ways

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.
1
vote
2answers
48 views

Sum of Binomial Series of form $\binom{2000}{3k-1}$

Find the Value of $$ \binom{2000}{2}+\binom{2000}{5}+\binom{2000}{8}+\cdots+\binom{2000}{1997}+\binom{2000}{2000}$$
2
votes
1answer
72 views

Using combinatorial reasoning to show $n!=\binom{n}{0}D_n+\binom{n}{1}D_{n-1}+\dots+\binom{n}{n}D_0$

How can one use combinatorial reasoning to show that $$n!=\dbinom{n}{0}D_n+\dbinom{n}{1}D_{n-1}+\dbinom{n}{2}D_{n-2}+....+\dbinom{n}{n-1}D_1+\dbinom{n}{n}D_0$$ Now $D$ stands for deranged which is a ...
0
votes
1answer
201 views

Project Euler #453 confusion

So I decided to give a shot on the #453 project euler problem but there is something that confuses me with the numbers given. I decided to start by calculating the possible arrangements of 4 vertices ...
0
votes
2answers
45 views

Couple of Counting (how many ways) questions.

1.If I have a group of 10 seats reserved for people, and there are n=>10 total people, how many ways are there to choose who gets the 10 seats? for ex:If there was a definite number of people lets ...
1
vote
2answers
46 views

Find solutions of this equation

If $a+b+c+d = 30$ and $a,b,c,d$ lie between $0$ and $9$. How to find number of solutions of this equation.
2
votes
1answer
68 views

Combinatorics - finding coefficients when summing over permutations of permutations

I have $N$ 2-tuples. Each tuple* can either be up, in which case it has components $(a,b)$, or it can be down, in which case it has components $(c,d)$. Given that exactly $N_\mathrm{up}$ of these ...
3
votes
2answers
90 views

combinatorics - fixed point permutations

Simple question but I just need a little tip to finish it. we are given $A=\{1,2,3...,2n-1,2n\}$ the set of all integers between and including $1$ and $2n$. We are asked how many different ...
0
votes
1answer
70 views

A few questions relating to counting for midterm practise exam?

I'm doing some questions for my midterm practise exam (multiple choice) for discrete structures and would appreciate some help (My answer is bolded): Using the 26-letter alphabet {a,b,c,...,z}, how ...
2
votes
0answers
183 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
0
votes
1answer
70 views

Combinations and their sum with constraints

I have a number of books (n). They all have different a different thickness and mass. I know that there are (2^n)-1 combinations to place the books. The order of the books does not matter. However ...
1
vote
1answer
206 views

Theorem regarding greatest common divisor of certain Binomial coefficients.

Recently my friend asked following question- find the greatest common divisor of all binomial coefficient for a given n so the problem is in mathematical form ...
1
vote
1answer
96 views

Number of ways to distribute $N/2$ blue and $N/2$ red balls over $N$ positions

I have read this one: Permutations with identical items which sounds like a duplicate, but I couldn't easily apply the answer to my question (in other words, I didn't get the answer, sadly) So here ...
4
votes
3answers
1k views

Distributing identical objects to identical boxes

We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
6
votes
5answers
797 views

Simplify $\sum_{i=0}^n (i+1)\binom ni$

Simplifying this expression$$1\cdot\binom{n}{0}+ 2\cdot\binom{n}{1}+3\cdot\binom{n}{2}+ \cdots+(n+1)\cdot\binom{n}{n}= ?$$ $$\text{Hint: } \binom{n}{k}= \frac{n}{k}\cdot\binom{n-1}{k-1} $$
0
votes
2answers
35 views

Relation between binomials

how can I prove that the following relation is true: $$\binom{x-2}y+2\binom{x-2}{y-1}+\binom{x-2}{y-2}=\binom{x}y$$ Thank you for hints or references! Marted
1
vote
3answers
6k views

Number of subsets of a set having r elements

We have studied the standard way of ascertaining the total number of subsets of a set by using the concept of combinations ( or binomial coeffecients ). I came across an alternate derivation for this ...
1
vote
1answer
223 views

permutations and the binomial coefficient

I have seen several times the use of "n choose k" in the left side of the permutations formula. However, this expression is usually referred to be used with combinations. Not that this change when or ...
3
votes
0answers
124 views

Defining a signed involution

We define the displacement of $\pi$ as $\mathrm{disp}(\pi)=\sum_{i=1}^n|\pi(i)-i|$. I know that it's even. Could you help me to find a good signed involution of the set of permutation with ...
1
vote
0answers
185 views

Interdependent constraints combination problem

I am trying to solve the following combination problem. You have 4 knobs or levers that have maximum values, such as 0-20, 0-30, 0-50 and 0-100. Their total values must equal an amount, say 47. Their ...
4
votes
3answers
1k views

Combination problem with constraints

You have four containers and one pitcher of water that holds 100L. Each container has different capacities with maximums of, say...70L, 45L, 33L and 11L levels respectively. What is the formula that ...
3
votes
0answers
358 views

Multinomial Coefficients ! [closed]

I have come across a paper that has suggested a formula for "NUMBER OF MULTINOMIAL COEFFICIENTS NOT DIVISIBLE BY A PRIME"; but I don't understand the notation.Please help. The formula is: $ G(n,l,p)= ...
2
votes
2answers
2k views

How do I determine the possible number of combinations of two ordered sets?

I'm not quite sure what the mathematical term for what I'm asking is, so let me just describe what I'm trying to figure out. Let's say that I have two ordered sets of numbers {1, 2} and {3, 4}. I'm ...