Prof. Tim Gowers’ on recognising countable sets

3) Observe the identity of answers to Problem 1 and 2. If you think it is a mere coincidence, experiment with other numbers in place of 16 and 4. If you do not think it is a mere coincidence, explain it and go to Problem 4. If you cannot explain its identity, experiment with other numbers.

4) Express the number $N(n,p)$ of ways of obtaining a total of n by rolling p dice, as a certain coefficient in a suitable product of binomial expansion in powers of t. If you cannot do this problem, go to problem 3.

5) Verify that the number of increasing words of length 10 out of the alphabet $(a,b,c,d,e)$ with $a<b<c<d<e$ is the coefficient of $t^{10}$ in $(1-t)^{-5}$ . Try to explain why this is so.

6) A child has a store of toy letters consisting of 4 A's, 3 B's, and 2 E's. (6a) How many different increasing four-letter words (given A<B<E) can it make? The child does not worry about the meanings of the words. (6b) Show that there exists a bijection between the set of increasing four-letter words of all possible lengths and the set of terms of the expansion $(1+A+A^{2}+A^{3})(1+B+B^{2}+B^{3})(1+E+E^{2})$ . (6c) Can you obtain the answer to (6a) as certain numerical coefficient in a suitable expansion of products?

7) The Parliament of India, in which there are n parties represented, wants to form an all-party committee (meaning, a committee in which there is at least one member from each party) of $r \geq n$ members. Assuming (a) that there exist sufficient members available for the committee in each party and (b) that the members of each party are indistinguishable among themselves(!), find the number of distinct ways of forming the committee. Show that this number may be expressed as a suitable coefficient in the binomial expansion of $(1-t)^{n-r}$

8) Find the number N of permutations of the multiset $(a,a,b,b,b)$ taken three at a time. If you have to express N as the coefficient $t^{3}$ or $t^{3}/3!$ or $3!t^{3}$ (the choice is yours) in the expansion of one of
8a) $(1+t)^{2}(1+t)^{3}$
8b) $(1+t)^{-2}(1+t)^{-3}$
8c) $(1+t+t^{2})(1+t+t+t^{2})$
8d) $(1+t+\frac{t^{2}}{2!})(1+t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!})$

which one would you choose? Can you justify your choice without using the value of N?

9) A family of 3, another family of 2, and two bachelors go for a joy ride in a giant wheel in which there are three swings A, B, C. In how many ways can they be seated in the swings (assuming there are sufficient number of seats in each swing) if the families are to be together? List all the ways.

10) n lines in a plane are in general position, that is, no two are parallel and no three are concurrent. What is the number of regions into which they divide the plane?

11) If $(a_{n})$ is a sequence of numbers satisfying $a_{n}-na_{n-1}=-(a_{n-1}-(n-1)a_{n-2})$ , find $a_{n}$ , given that $a_{0}=1$ and $a_{1}=0$ .

12) If $a_{n}$ is the number of ways in which we can place parentheses to multiply the n numbers $x_{1}, x_{2}, \ldots, x_{n}$ on a calculator, find $a_{n}$ in terms of the $a_{k}$ ‘s where $k=1,2, \ldots, (n-1)$ .

13) A graph $G=G(V,E)$ is a set V of vertices a, b, c, $\ldots$ , together with a set $E=V \times V$ of edges $(a,b), (a,a), (b,a), (c,b), \ldots$ . If $(x,y)$ is considered the same as $(y,x)$ , we say this graph is undirected. Otherwise, the graph is said to be directed, and we say ‘(a,b) has a direction from a to b’. The edge $(x,x)$ is called a loop. The graph is said to be of order $|V|$ .

If the edge-set E is allowed to be a multiset, that is, if an edge $(a,b)$ is allowed to occur more than once (and this may be called a ‘multiple edge’), we refer to the graph as a general graph.

$\phi_{5}(n)$ and $\psi_{5}(n)$ denote the numbers of undirected (respectively,directed) loopless graphs of order 5, with n edges, none of them a multiple edge, find the series $\sum \phi_{5}(n) t^{n}$ and $\sum \psi_{5}(n) t^{n}$ .

Cheers,
Nalin Pithwa.

Tutorial problems for RMO 2019 : combinatorics continued

Posted by Nalin Pithwa

1) In how many ways can 5 men and 5 women be seated in a round table if no two women may be seated side by side?

2) Six generals propose locking a safe containing top secret with a number of different locks. Each general will be given keys to certain of these locks. How many locks are required and how many keys must each general have so that, unless at least four generals are present, the safe cannot be opened?

3) How many integers between 1000 and 9999 inclusive have distinct digits? Of these, how many are even numbers? How many consist entirely of odd digits?

4) In how many ways can 9 distinct objects be placed in 5 distinct boxes in such a way that 3 of these boxes would be occupied and 2 would be empty?

5) In how many permutations of the word AUROBIND do the vowels appear in the alphabetical order?

6) There is an unlimited supply of weights of integral numbers of grams. Using n or fewer weights, find the number of ways in which a weight of m grams can be obtained. Prove that there is a bijection of the set of all such ways on the set of increasing words of length $(n-1)$ or $(m+1)$ ordered letters.

7) How many distinct solutions are there of $x+y+z+w=10$ (a) in positive integers and (b) in non-negative integers?

8) A train with n passengers aboard makes m stops. In how many ways can the passengers distribute themselves among these m stops as alighting passengers? if we are concerned only with the number of alighting passengers at each stop, how would the answer be modified?

9) There are 16 books on a bookshelf. In how many ways can 6 of these books be selected if a selection must not include two neighbouring books?

10) Show that there are ${(n=5)} \choose 5$ distinct throws of a throw with n non-distinct dice.

11) Given n indistinguishable objects and n additional distinct objects —- also distinct from the earlier n objects — in how many ways can we choose n out of the 2n objects?

12) Establish the following relations:
12a) $B_{n+1}=\sum_{k=0}^{n}(B_{k}){n \choose k}$
12b) $\sum_{k}{p \choose k}{q \choose {n-k}}={{p+q} \choose n}$
12c) $S_{n+1}^{m} = \sum_{k=0}^{n}{n \choose k}S_{k}^{m-1}$
12d) $n^{p}=\sum_{k=0}^{n}{n \choose k}k! (S_{p}^{k})$

13) Prove the following identity for all real numbers x:
$x^{n}= \sum_{k=1}^{n}S_{n}^{k}[x]_{k}$

14) Express $x^{4}$ in terms of ${x \choose 4}$ , ${x \choose 3}$ , …by using the $S_{n}^{k}$ ‘s. Express ${x \choose 4}$ in terms of $x^{4}$ , $x^{3}$ , …by using the $s_{n}^{k}$ ‘s.

15) A circular loop is divided into p parts, p prime. In how many ways can we paint the loop with n colours if we do not distinguish between patterns which differ only by a rotation of the loop? Deduce Fermat’s Little theorem: $n^{p}-n$ is divisible by p if p is prime.

16) In problem 15, prove that $n^{p}-n$ is also divisible by 2p if $p \neq 2$ . Where is the hypothesis that p is prime used in Problem 15 or in this problem?

17) How many equivalence relations are possible on an n-set?

18) The complete homogeneous symmetric function of n variables $\alpha_{1}$ , $\alpha_{2}$ , $\ldots$ , $\alpha_{n}$ of degree r is defined as $h_{r}(\alpha_{1},\alpha_{2}, \alpha_{3}, \ldots, \alpha_{n})=\sum \alpha_{1}^{i_{1}}\alpha_{2}^{i_{2}}\ldots \alpha_{n}^{i_{n}}$ the summation being taken over all ordered partitions of r, where the parts are also allowed to be zero. How many terms are there in $h_{r}$ ?

Test yourself ! Improve your mettle in math !
Regards,
Nalin Pithwa.

Pre RMO or PRMO problem set in elementary combinatorics

Posted by Nalin Pithwa

1) How many maps are there from an n-set to an m-set? How many of these are onto? How many are one-one? Under what conditions?

2) Consider the letters of the word DELHI. Let us form new words, whether or not meaningful, using these letters. The *length* of a word is the number of letters in it, e.g., the length of “Delhi” is 5; the length of “Hill” is 4. Answer the following questions when (a) repetition of letters is not allowed and (b) repetition of the letters is allowed:
(i) How many words can be formed of length 1,2,3,4,…?
(ii) How many words in (i) will consist of all the letters?
(iii) How many words of the words in (i) will consist of 1,2,3,4, …specified letters?
(iv) How many of the words in (i) will consist of only 1,2,3,4,…letters?
(v) How many of the words in (i) will be in the alphabetical order of the letters?

3) Repeat Problem 2 with the word MISSISSIPPI.

4) Suppose there are 5 distinct boxes and we want to sort out 1,2,3,…,n objects into these boxes.
4i) In how many ways can this be done?
4ii) In how many of these situations would no box be empty?
4iii) In how many of the above would only 1,2,3,4 … specified boxes be occupied?
4iv) In how many would only 1,2,3,4…boxes be occupied?
4iv) If the objects are indistinguishable from one another, how would the answers to (i) to (iv) change, it at all?

If there is an added restriction that each box can hold only one object and no more, what will be the answers to (i) to (v)?

5) Repeat Problem 4 with 9 boxes.

6) Repeat Problem 4 with 5 non-distinct (=indistinguishable identical) boxes.

7) Repeat Problem 6 with 9 boxes.

8) How many 5-letter words of binary digits are there?

9) Ten teams participate in a tournament. The first team is awarded a gold medal, the second a silver medal, and the third a bronze medal. In how many ways can the medals be distributed?

10) The RBI prints currency notes in denominations of One Rupee, Two Rupees, Five Rupees, Ten Rupees, Twenty Rupees, Fifty Rupees, and One hundred rupees. In how many ways can it display 10 currency notes, not necessarily of different denominations? How many of these will have all denominations?

11) In how many ways can an employer distribute INR 100/- as Holiday Bonus to his 5 employees? No fraction of a rupee is allowed. Also, do not worry about question of equity and fairness!

12) The results of 20 chess games (win, lose, or draw) have to be predicted. How many different forecasts can contain exactly 15 correct results?

13) How many distinct results can we obtain from one throw of four dice? five dice? Can you generalize this?

14) In how many ways can 8 rooks be placed on a standard chess board so that no rook can attack another? How many if the rooks are labelled? How would the answer be modified if we remove the restriction that “no rook can attack another”?

15) Show that there are 7 partitions of the integer 5, and 33 partitions of the integer 9. How many of these have 4 parts ? How many have the largest part equal to 4? Experiment with other partitions and other numbers.

Cheers,
Nalin Pithwa.

William Lowell Putnam 2018 results

Posted by Nalin Pithwa

https://www.ams.org/news?news_id=4936

Hats off to all the winners !

RMO and Pre RMO Geometry Tutorial Worksheet 1: Based on Geometric Refresher

Posted by Nalin Pithwa

1) Show that quadrilateral ABCD can be inscribed in a circle iff $\angle B$ and $\angle D$ are supplementary.

2) Prove that a parallelogram having perpendicular diagonals is a rhombus.

3) Prove that a parallelogram with equal diagonals is a rectangle.

4) Show that the diagonals of an isosceles trapezoid are equal.

5) A straight line cuts two concentric circles in points A, B, C and D in that order. AE and BF are parallel chords, one in each circle. If CG is perpendicular to BF and DH is perpendicular to AE, prove that $GF = HE$ .

6) Construct triangle ABC, given angle A, side AC and the radius r of the inscribed circle. Justify your construction.

7) Let a triangle ABC be right angled at C. The internal bisectors of angle A and angle B meet BC and CA at P and Q respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.

8) Three circles $C_{1}, C_{2}, C_{3}$ with radii $r_{1}, r_{2}, r_{3}$ , with $r_{1}<r_{2}<r_{3}$ . They are placed such that $C_{2}$ lies to the right of $C_{1}$ and touches it externally; $C_{3}$ lies to the right of $C_{2}$ and touches it externally. Further, there exist two straight lines each of which is a direct common tangent simultaneously to all the three circles. Find $r_{2}$ in terms of $r_{1}$ and $r_{3}$ .

Cheers,

Nalin Pithwa