b.

a.

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2

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In how many ways can the letters of the word "SOLUTION" be arranged so that the

vowels always come together?

(2)

A tennis player has three weeks to prepare for a tennis tournament. She decides to play
at least one set every day but not more than 36 in all. Show that there is a period of

consecutive days during which she will play exactly 21 sets.

(3)

It is estimated that 50% of emails are spam emails. Some software has been applied to
filter these spam emails before they reach your inbox. A certain brand of software
claims that it can detect 99% of spam emails, and the probability for a false positive (a
non-spam email detected as spam) is 5%. Now if an email is detected as spam, then

what is the probability that it is in fact a non-spam email?

X is a normally distributed variable with mean ‏پر‎ = 30 and standard deviation

iFind i) P(Q0
(4)
64.
(5)

There are 8 guests in a party. Each guest brings a gift and receives another gift in
return. No one is allowed to receive the gift they bought. How many ways are to

distribute the gifts ?
101, 99

(2)

Find the coefficient of x° in the expansion of (2-5) 10 x y in the expansion of

(2x-3yy?_.
Find the mean of Poisson distribution.

PART C
State and prove Lagrange’s theorem.

Describe Planar, Euler and Hamiltonian graph.

(3)
(4)

(6)
(6)

Define a subgroup. Prove that the necessary & sufficient condition that a non-empty

subset H of a Group G be a Subgroup is ೩ €H, 0 €H => எ.
Explain Warshalls algorithm.

What is a decision tree?
State and prove 5-color theorem.
Prove that Zn= {0,1,2,...n-1} is an abelian group under addition modulo n.

What is meant by discrete logarithm?

(6)
(4)
(2)
(6)
(3)
(3)