All Questions
68
questions
1
vote
0
answers
64
views
How many permutations $\left(a_1,a_2,\dots,a_n \right)$ of $\left(1,2,\dots,n \right)$ such that...
Source
How many permutations $\left(a_1,a_2,\dots,a_n \right)$ of $\left(1,2,\dots,n \right)$ such that $$\left|a_{i+1} - a_i \right| \le 2, \ \forall i=1,2,\dots,n-1.$$
At first I thought this might ...
- 31
-2
votes
2
answers
141
views
Probability for finding product?
A number is chosen randomly from the first billion natural numbers.
What is the probability that the product of the number with its two immediate successors is divisible by 24 ?
- 3
2
votes
0
answers
35
views
Linear extension of a divisors set
For a number $N$, let $S_N$ be its set of divisors, and let $C(N)$ be the number of arrangements of $S_N$ in which every divisor itself appears after all of its divisors.
$C(12)=5$, because of the ...
- 1,022
2
votes
2
answers
141
views
Each row of an n x n table form an arithmetic progression. Find n such that table can be transformed so that each column form arithmetic series
We place $ 1,\ldots,n^2$ integer numbers into an $n \times n$ table. We call this table good if each row can be permuted to form an arithmetic progression. For what value of $n$ can we transform (by ...
XMannnn
0
votes
0
answers
84
views
Consider the increasing list of positive integers that do not contain the digit zero, 1, 2, 3... 9, 11, 12,.... The 2022nd number in this sequence is
Consider the increasing list of positive integers that do not contain the digit zero, $1, 2, 3, \ldots, 9, 11, 12, \ldots$. The $2022$nd number in this sequence is
so the number of single-digit ...
0
votes
0
answers
82
views
cards permutations
This Question is from HackerRank : Alice was given the $n$ integers from $1$ to $n$ . She wrote all possible permutations in increasing lexicographical order, and wrote each permutation in a new line. ...
user411780
3
votes
5
answers
825
views
How many whole numbers between $100$ and $800$ contain the digit $2$?
I had a very strange doubt in this question while I was solving it. Now in order to solve first I calculated the three digit numbers which won't have $2$ at all in them and the number of such three ...
- 1,699
0
votes
0
answers
64
views
Closed form solution of this Vandermonde-like Identity
Can a closed form solution for the following expression be obtained? ($\alpha, n, m$ are constants)
$$
\sum_{i=0}^k\alpha^i \binom{n}{i}\binom{m}{k-i}
$$
I know that $\binom{n+m}{k} = \sum_{i=0}^k\...
- 289
2
votes
1
answer
290
views
Asif , Aniqa and Saki are perplexed with grid and numbers
Asif has filled in a 3×3 grid with the numbers 1,2, . . . ,9. Saki writes down the three numbers obtained by multiplying the numbers in each horizontal row. Aniqa writes down the three numbers ...
- 479
2
votes
3
answers
103
views
Combinatorics, counting and number theory.
Find number of ordered quadruples (a,b,c,d) [of positive integers] such that $lcm[a,b,c]= lcm[a,b,d]= lcm[a,c,d]= lcm[b,c,d] = 2^r\cdot3^s$
So i approached it like two of a,b,c,d has max power of 2 = ...
- 1,863
0
votes
1
answer
262
views
How do you prove mathematically this four card trick?
Shuffle a deck of 4 cards. Ask the participant to remember the bottom card. Move top card to bottom. Flip the first card. Cut (not shuffle) the 4 cards in any way (or not at all). Do a two card flip/...
- 1,958
1
vote
1
answer
65
views
Find The sum of all number in $\mathrm{S}$ is?
Let $S$ be the set of all 3 - digits numbers. Such that
(i) The digits in each number are all from the set $\{1,2,3, \ldots ., 9\}$
(ii) Exactly one digit in each number is even
The sum of all number ...
user791682
0
votes
1
answer
54
views
Number of ways to colour number line.
I thought of this question:
Consider you have a number line that starts at $0$ and ends at $n$ where $n>0$. Consider there are $k$ number of points on this number line which are said to be lucky,...
-1
votes
2
answers
73
views
Does there exists such quadruple's of positive integers? [closed]
Problem: Let $S=\{1,2,...,8\}$ be the set consisting first eight positive integer from which each integer present can be taken twice not more than that. Select a quadruple $(a,b,c,d)$ from $S$, where $...
- 2,318
-1
votes
1
answer
41
views
Question on finding sum of all numbers formed by given numbers where zero is also included.
Find the sum of all the odd numbers of five-digit that can be made using the digits $0,1,4,5,4.$
- 135
3
votes
1
answer
720
views
How many non-negative integer solutions exist for: $x+y+z=48$ where, $x<y<z$?
I want to find the number of non-negative integral solutions to the following:
$x+y+z=48$ where, $x<y<z$.
The answer is apparently 192 and the solution provided is $$\frac{\dbinom{50}{2}-\...
- 123
0
votes
3
answers
213
views
In how many ways can we form a committee of positive size from $7$ women, $4$ men so that there are at least $2$ women in the committe?
In how many ways can we form a committee of positive size from $7$ women, $4$ men so that there are at least $2$ women in the committee?
So the committee size must be $\geq 2$ and $\leq 11$ since it ...
- 2,533
4
votes
1
answer
244
views
Why does this appear to produce OEIS sequence A263484?
A263484 is "Triangle read by rows: $T(n,k)$ ($n\geq 1$, $0 \leq k < n$) is the number of permutations of $n$ with $n! - k$ permutations in its connectivity set.", and the sequence is:
1,
1,1,
1,...
- 351
1
vote
3
answers
151
views
In determining probability using 2 dice rolls why are permutations (x,x) not counted twice?
So I've been working in probability regarding dice rolls. I came across this problem:
If you roll 2 dice, what is the probability the first die is a 6 given that you rolled an 8?
This is clearly a ...
- 497
3
votes
2
answers
1k
views
Number of six digit numbers divisible by $3$ but none of the digits is $3$
Find number of six digit numbers divisible by $3$ but none of the digits is $3$
My try:
Let the six digits are $a,b,c,d,e,f$ such that
$$a+b+c+d+e+f=3p$$
where $1 \le p \le 18$
Now since $a \ge 1$...
- 10.2k
2
votes
3
answers
62
views
How can I increase the complexity of a number and maintain uniqueness
I have an 8-digit number and you have an 8-digit number - I want to see if our numbers are the same without either of us passing the other our actual number. Hashing the numbers is the obvious ...
1
vote
1
answer
105
views
Permutations, products, and unit fractions
Here's a question motivated by some related MathOverflow questions of Zhi-Wei Sun.
Show that, for any $n \ge 1$, there is a permutation of $\{1,2,\ldots, n\}$, i.e., some $\pi \in S_n$, such that $$\...
- 2,322
1
vote
1
answer
141
views
Counting Credit Cards
The credit cards (VISA CARDS & MASTER CARDS) numbers have the following properties;
Let $N$ be the card number
$N$ has $16$ digits; $N = a_{1}a_{2}a_{3}...a_{16}$ where $a_{k}$ is the $k$-th ...
- 5,450
2
votes
0
answers
52
views
How to minimize the cardinality of the intersection of two sets of pairs from a cyclic group?
I was in a group of 10 persons and we wanted to make a know-each-other-better game, so we did the following thing: 5 of us were standing in the middle (inner circle) and the other five were standing ...
- 121
0
votes
2
answers
204
views
Show that $\binom {2n}n \leq(2n)^{\pi (2n)}$ where $\pi(2n) $ is number of prime number less than $2n$
Show that $$\binom {2n}n \leq(2n)^{\pi (2n)}$$
where, as usual, $\pi(2n) $ is number of prime number less than $2n$.
I was solving basic techniques of combinatorial theory by Daniel Cohen.
I was ...
- 103
-3
votes
3
answers
69
views
How many numbers can be formed? [closed]
How many numbers can be formed from 1, 2, 3, 4, 5, ( without repetition), when the digit at the unit's place must be greater than that in the ten's place?
-3
votes
2
answers
89
views
24!=a!*b!*c!*d! Find total number of ordered quadrapules which satisfy the given equation? [closed]
I have found 92 as answer after trying possible quadrapules.
Can anyone suggest different approach?
0
votes
0
answers
44
views
Equation for non-cyclic sequence based around pascal's triangle
This question is an extension of Need help in determining where this pascal's triangle-like sequence comes from.
To summarise, a binary sequence of length n ...
- 237
2
votes
3
answers
11k
views
The number of ways in which one can choose three distinct numbers from the set so that the product of the chosen numbers is divisible by $9$
Consider the set $A=\{1,2,3,...,30\}$
. The number of ways in which one can choose three
distinct numbers from $A$ so that the product of the chosen numbers is divisible by $9$ is $X$
Find $X$...
- 2,132
5
votes
2
answers
4k
views
Maximum possible order of an element in $S_7 \text{ and } S_{10}$
Exercise :
Find the maximum possible order of an element of the group of permutations $S_7$. Do the same thing for $S_{10}$.
Discussion :
Recalling that any permutation can be written as a ...
- 21.4k