All Questions
Tagged with algebra-precalculus combinatorics
739
questions
1
vote
1
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98
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Two numbers written on a board get replaced
Question: "Several (at least two) nonzero numbers are written on a board. One
may erase any two numbers, say $a$ and $b$, and then write the numbers
$a+\frac{b}{2}$
and $b−\frac{a}{2}$
instead. ...
- 123
1
vote
1
answer
37
views
Counting Matrices: Comparing two different approaches
I posted a question on this, and the number of matrices came to be 36:
The number of $3\times3$ non singular matrices with four entries as 1 and all other entries as 0, is (JEE Main 2010)
However, I ...
- 1,834
0
votes
0
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20
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Bijection between relation and matrix
The number of $3\times3$ non singular matrices with four entries as 1 and all other entries as 0, is [(JEE Main 2010)]
Inspired by this question (on representing relations using matrices), I tried ...
- 1,834
4
votes
1
answer
173
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$10$-digit numbers divisible by $66667$.
Let us consider the set $S$ of $10$-digit numbers whose digits are only allowed to be taken from the set $\{3,4,5,6,7,8\}$. Let $S'$ be the subset of $S$ such that its elements are divisible by $66667$...
- 97
5
votes
1
answer
73
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Counting non singular matrices
The number of $3\times3$ non singular matrices with four entries as 1 and all other entries as 0, is (JEE Main 2010)
My approach: the determinant must be non-zero, which means that no row or column ...
- 1,834
2
votes
1
answer
63
views
Tournament between 10 players, maximum number of games, also minimum number of wins to get 4th place
Say we have a video game tournament, in which 10 gamers play all against each other. Assume that each match ends in one person winning and another losing, no draws.
What is the maximum number of ...
- 337
0
votes
3
answers
141
views
The 50 game between two players, selecting numbers between 1 and 10 inclusive + variations
Let's play a game with two players, with player 1 going first. The players take turns selecting a number between 1 and 10 inclusive. The person who says the number that makes the sum reach or exceed ...
- 393
2
votes
1
answer
172
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Let $x,y,z$ are selected from set of natural numbers. Find the probability that $x^2+y^2+z^2$ is divisible by $5$
Let $x,y,z$ are selected from set of natural numbers. Find the probability that $x^2+y^2+z^2$ is divisible by $5$
My Method: I took set of first $10$ natural numbers. Let say set is $S=\{1,2,3,...,10\...
- 3,835
2
votes
2
answers
76
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In a tournament, there are twelve players $S_1,S_2,....,S_{12}$ and divided into six pairs at random
In a tournament, there are twelve players $S_1,S_2,....,S_{12}$ and divided into six pairs at random. From each game a winner is decided on the basis of a game played between the two players of the ...
- 3,835
3
votes
5
answers
217
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Coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$
Find the coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$
I tried splitting the terms inside the bracket into two parts $1+x+\dots+x^9$ and $x^{10}$, and then tried binomial theorem, but that ...
- 545
2
votes
3
answers
119
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Probability of crossing value $N= 1000$ with face $k$ when adding up 6-sided dice rolls? Also two 6-sided dice?
We roll a 6-side die repeatedly until the accumulated sum exceeds $N=1000$. What is the probability that the last roll (last face) equals $k$?
I've tried doing this by hand: We go over 1000 with 1 by ...
- 337
0
votes
3
answers
109
views
A person tosses a fair coin n times, they win if they are able to get heads in multiple of 3
A person tosses a fair coin $n$ times, they win if they are able to get heads in multiple of $3$. The probability they win given that they get heads at least one time is ($n$ is not multiple of $3$ ...
- 3,835
0
votes
1
answer
39
views
Lower bound on sequence
I have the following question from my textbook:
For any integer $k \geq 2$, prove that if $A$ the number of integers $\leq x$ is $y$ such that $y$ can be written as the sum of $k$ many $k$-th powers ...
- 13
2
votes
3
answers
62
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Verification of answer in a birthday problem
In the answer here, should the number of ways to pick the groups of triples, pairs and singlets from the 20 people be:
$$\frac{20!}{\color{#C00}{3!^2}\,\color{#090}{2!^4}\,\color{#E90}{6!}}$$
since if ...
- 1,834
-3
votes
1
answer
54
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How to set equations containing factorials? [closed]
I recently encountered a problem involving the construction of equations involving factorials or combinatorial numbers. I recall reading somewhere (although I cannot recall the reference) that this ...
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