5.

௦. Itis known that the probability that a recovered fossil human skull being (3) female
is 0.6 . What is the probability that out of 6 skulls, exactly 4, will be female?

a. Eight coins are thrown simultaneously. Using Binomial theorem, find the (3) chance of

obtaining atleast 6 heads.

b. In how many ways can the letters of the word ALLAHABAD be arranged ? (3) How many
of these permutations are there in which
(i) Two L's come together? (ii) Two L's donot come together ?

௦. Twodiceare rolled. if the first one top with 5, then find the probability that the total ofthe

two will be greater than 7.
. Find the expected value ofa Uniform Random Variable. (3)

ರ ©

Find the coefficient of in the expansion of (2x-y-z) 1 910 xyz-2 in the (3) expansion
of (x-2Y+3zel )
c. The CEO's of 18 software companies meet to discuss a problem. In how many (3) ways

can they sit themselves around a table so that CEO of GE, MCL and Infosys choose to sit
together.

PART C

a. 1113 Ring 'R' with Unity (xy)? = xy? for all x, ye र then रि is Commutative. (4)

b. Prove that " If a graph (connected or disconnected )has exactly two vertices of (4) odd
degree, there must be a path joining these two vertices". c What is meant by Discrete
Logarithms ?

aProve that a Subgroup H of a group © is Normal if and only if 8-1 hge ಗ, for (4) every

hEH, gEG.

b. Discuss Residue arithmetic. (4)

©. Prove that a graph with atleast one edge is 2-chromatic if and only if it has no (4) circuits
ofodd length.

a. Show that the Graph (65 is not Coplanar.

0. Prove that the necessary & sufficient condition that a non-empty subset 'H' of a Group '6'
be a Subgroup is a EH, b Eli abei EH.

c. Discuss Elliptic Curve Arithmetic. (4)