 # Understand the problem

Suppose that in a sports tournament featuring n players, each pair
plays one game and there is always a winner and a loser (no draws).
Show that the players can be arranged in an order P1, P2, . . . , Pn such
that player Pi has beaten Pi+1 for all i = 1, 2, . . . , n.

##### Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016. Subjective Problem no.1.

Combinatorics

7 out of 10

##### Suggested Book

Do you really need a hint? Try it first!

we define a sequence $p_1$>$p_2$>$p_3$………..>$p_n$ . such that $p_i>p_{i+1}$ implies $p_i$ has beaten $p_{i+1}$

let assume that the statement is true for n=k  i.e $p_1> p_2>p_3………p_k$ now look into the case of $p_{k+1}$

if $p_{k+1}$ > $p_1$ or $p_k$>$p_{k+1}$ we are done (why?)

if $p_k$<$p_{k+1}$ or $p_{k+1}$<$p_1$ then  we can find out at least one pair $p_i$ >$p_{i+1}$ such that $p_i$> $p_{k+1}$ >$p_{i+1}$ , we can just find it by shifting $p_{k+1}$ term by term [ you can make some small experiment ]. other wise it contradict our initial assumption that $p_k$<$p_{k+1}$ or $p_{k+1}$<$p_1$ [ why?]   hence we are done

# Connected Program at Cheenta # I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

# Similar Problem

## ISI MStat Entrance 2020 Problems and Solutions

Problems and Solutions of ISI MStat Entrance 2020 of Indian Statistical Institute.

## ISI Entrance 2020 Problems and Solutions – B.Stat & B.Math

Problems and Solutions of ISI BStat and BMath Entrance 2020 of Indian Statistical Institute.

## Testing of Hypothesis | ISI MStat 2016 PSB Problem 9

This is a problem from the ISI MStat Entrance Examination,2016 making us realize the beautiful connection between exponential and geometric distribution and a smooth application of Central Limit Theorem.

## ISI MStat PSB 2006 Problem 8 | Bernoullian Beauty

This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

## How to roll a Dice by tossing a Coin ? Cheenta Statistics Department

How can you roll a dice by tossing a coin? Can you use your probability knowledge? Use your conditioning skills.

## ISI MStat PSB 2009 Problem 8 | How big is the Mean?

This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

## ISI MStat PSB 2009 Problem 4 | Polarized to Normal

This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 4. It is based on the idea of Polar Transformations, but need a good deal of observation o realize that. Give it a Try it !

## ISI MStat PSB 2008 Problem 7 | Finding the Distribution of a Random Variable

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 7 based on finding the distribution of a random variable. Let’s give it a try !!

## ISI MStat PSB 2008 Problem 2 | Definite integral as the limit of the Riemann sum

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 2 based on definite integral as the limit of the Riemann sum . Let’s give it a try !!

## ISI MStat PSB 2008 Problem 3 | Functional equation

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 3 based on Functional equation . Let’s give it a try !!