A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green?
See Answer →Show that if 7 colours are used to paint 50 bicycles and each bicycle is coloured with a single colour, at least 8 bicycles will have the same colour.
See Answer → is Hamiltonian, how are m and n related? Justify your answer
Find a recurrence relation for an , the number of ways to arrange cars in a row with n spaces if we can use Maruti 800, Tata Safari or Scorpio. A Tata Safari or Scorpio requires two spaces, whereas a Maruti 800 requires just one space. Assume that you have unlimited number of each type of car and we do not distinguish between 2 cars of the same type.
See Answer →Express 5 x in terms of falling factorials and hence evaluate m S5 for m = 0,1,2,3,4,5.
See Answer →write down and count all the partitions of the number 7. To verify your answer use the generating function for , Pn taking n= 7 in Theorem 5 (of Unit 5, Block2).
See Answer →In the June, 2021 Term-End Examination of MTE-13, it was asked to give a direct and an indirect proof of the following statement.
such that a is even and a +b is even, then b is even."
One student gave an indirect proof as follows:
“Let b be m +1, which is an odd number. We already know a and a +b are even. If we substitute b + m +1 in a + b, it becomes a + m +1, which is an odd number. This contradicts the given statement. Hence b is an even number.”
What is wrong with the above proof ? Also give a correct direct and an indirect proof.
See Answer →What do you understand by a subdivision of a graph? Is every subdivision of a Hamiltonian graph Hamiltonian? Justify.
See Answer →Is the complement of the Peterson graph planar? Justify your answer.
See Answer →Is the complement of the Peterson graph planar? Justify your answer.b) Is the complement of the Peterson graph planar? Justify your answer.
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Draw three nonisomorphic induced subgraphs of the following graph, each having the same number of vertices. Justify your choice.
See Answer →For any statements p,q and r, prove that
List all the onto mappings from the set {a,b,c,d} to {1,2,3,4}. How many onto
mappings are there form {a,b,c,d} to {1,2,3,4,5}?
See Answer →There are about 77 crore ways to arrange the letters of the word “COMBINATORICS”. Count the exact number of such ways.
See Answer →Using generating functions find
Express the following statements in symbolic form.
i) There is a man in the park with blue eyes.
ii) Every blue-eyed man in the park is wearing a red hat.
iii) If a man wears no hat, then he has black eyes
See Answer →Draw the logic circuit for the Boolean expression
Check whether the following statements are true or not. Justify your answers with a short proof or a counter example
i) If the contrapositive of a statement is true, then the statement itself is also true.
ii) is a linear homogeneous recurrence relation.
iii) A particular solution of the recurrence relationhas the form
iv) There exists a boolean expression in variables
v) If a dice is rolled thrice, then the probability of getting a 6 each time is
vi) Every odd cycle has the same chromatic and edge chromatic numbers.
vii) Every Eulerian graph is Hamiltonian.
viii) 3 4 S gives the number of ways in which any 3 objects can be placed in any 4 boxes.
ix) There exists a self-complementary planar graph on 5 or more vertices.
x) The number of partitions of 6 is 10.
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