All Questions
68
questions
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132
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Given a number m with n digits, find the sum of all permutations of m with exactly n digits
I've been struggling to find this solution. So far, I know the sum of all permutations of $m$, for example $123$, is $2!(1+2+3)(111) = 1332$.
Also I got that the number of permutations of $m$ with no ...
1
vote
1
answer
102
views
The frequency of a sum ended with specific decimal numbers
Randomly pick $\color{red}5$ distinct integers from $1, 2,3,4, 5,\cdots, 76, 77,78,79,\color{red}{80}$, and then sum the five integers together.
Try to find the frequencies when the last decimal ...
- 2,092
3
votes
4
answers
2k
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Number of ways of splitting $2310$ as product of three factors
Find Number of ways of splitting $2310$ as product of three factors.
My Try:
$$N=2310=3 \times 7 \times 2 \times 5 \times 11$$
$1.$ if two of the factors are ones ten trivially it is one way.
$2.$...
- 13.2k
3
votes
5
answers
3k
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Calculate the number of ways to paint $5$ buildings with $4$ colours such that all $4$ colours must be used
A developer has recently completed a condominium project in a valley. There are blocks of buildings $A$, $B$, $C$, $D$ and $E$ as shown in the diagram below.
The developer has colours ...
- 15.9k
5
votes
0
answers
193
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Number of integers containing every digit atleast once [duplicate]
I have to find out number of integers that have every digit $(0-9)$ in it where integer starting with zero doesn't count i.e leading zero shouldn't be there.
So this makes the length of an integer $ \...
- 51
1
vote
1
answer
116
views
Who will win the Matrix game?
I have a matrix of size $N \times M$, There is a doll placed at $(1,1)$, which is the upper-left corner of the matrix. Two player makes an alternative turn.
If the doll is at $(x,y)$ then a player ...
- 340
1
vote
1
answer
2k
views
Increasing or decreasing permutation of $N$ numbers
Given $N$ numbers, how many ways can we arrange those numbers such that they are Strictly increasing or decreasing.
For eg - $N=2$ we have $1,2$ so possible rearrangements are-
{{1,2} , {2,1}}
Can ...
- 53
2
votes
0
answers
514
views
Number of arrays with adjacent elements coprime?
An array of integers is called m-coprime if the following conditions are both satisfied:
All the integers in the array are positive divisors of m.
Each pair of adjacent elements in the array is ...
- 29
1
vote
1
answer
25
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$^nC_r=\sum_{i=0}^k a_i\,^kC_i$
$k,r,n$ are positive integer such that $1<k<r<n$. Find $a_i,0\le i\le k$ such that:
$^nC_r=\sum_{i=0}^k a_i\,^kC_i$
- 183
1
vote
2
answers
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How many words, with or without meaning, each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE? [duplicate]
My solution:
In the word INVOLUTE, there are $4$ vowels, namely, I,O,E,U and $4$
consonants, namely, N, V, L and T.
The number of ways of selecting $3$ vowels out of $4 = C(4,3) = 4$.
The number of ...
- 214
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 $N = 2$,...
- 619
0
votes
0
answers
620
views
Number of distinct integer-valued vector solution for $x_1 + x_2 + ... + x_r = n$ [duplicate]
The Number Of Integer Solutions Of Equations $$x_1 + x_2 + ... + x_r = n$$
An approach is to find the number of distinct non-negative integer-valued vectors $(x_1,x_2,...,x_r)$ such that $$x_1 + x_2 +...
- 101
0
votes
0
answers
24
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Find possible number of lists that can be formed. [duplicate]
I am new to such problems of number theory. Any help will be appreciated.
I have a list containing n numbers. I can apply the following operation exactly K times.
Pick some element in the array and ...
- 25
0
votes
1
answer
642
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a[i] denote number of friends i-th student has. c[j] denote frequency having at least j friends. Show that: ∑a[i]=∑c[j]. [duplicate]
Q. A class has 100 students. Let a[i], 1≤i≤100, denote the number of friends the i-th student has in the class. For each 0≤j≤99, let c[j] denote the number of students having at least j friends. Show ...
- 419
0
votes
1
answer
1k
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Number of valid parenthesis
I have to find out the number of valid parenthesis.Parenthesis are of two type [] ,(). How many ways are there to construct a valid sequence using ...
- 53