All Questions
68
questions
51
votes
1
answer
1k
views
How many ways can I arrange the numbers $1$ to $N$ with this divisibility condition?
For the numbers $1, \ldots, N$, how many ways can I arrange them such that either:
The number at $i$ is evenly divisible by $i$, or
$i$ is evenly divisible by the number at $i$.
Example: for $N = 2$,...
- 1,745
8
votes
2
answers
697
views
How many numbers of length N(N is the number of digits) are possible
Given $N \le 10^9$, determine how many numbers of length $N$ are possible with the following constraint: the adjacent digits of the number should have an absolute difference of $1$.
For eg, for $N = ...
- 10.3k
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...
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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 ?
- 7,388
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 ...
- 111
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 ...
- 77k
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. ...
CommunityBot
- 1
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 ...
- 7,388
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 ...
- 12.7k
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 = ...
- 126k
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/...
- 13.8k
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 ...
- 65.2k
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,...
- 620k