4(a) Five friends run a race everyday for 4 months (excluding Feb), If no race ends
in a tie, Show that there are at least 2 races with identical outcomes. (3)

(b) In how many ways can (i) 12 balloons be distributed .at a birthday party among
10 children (ii) Find the number of ways the balloons can be distributed if we
ensure that every child gets at least one balloon.

(3)
(c)Discuss various methods of enumeration. (3)
5(a) List out the important properties of Pascal's Triangle. (3)

(b) Determine the coefficient of x9y3 in the expansion of (x+y) ۶, (x+2y) ೩೧6

(2x-3y)12 using binomial theorem. (3)

(c) Explain Binomial distribution. A fair coin is tossed 50 times. What is the
probability that heads will appear exactly 25 times? (3)

6(a) 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 there to distribute the gifts? http:/GMvw.ktuonline.com (3)

(b) Find the mean & variance of a Poisson distribution. (3)

(c)Suppose an Item is manufactured by three machines X, Y, Z. All the three
machines have equal capacity and are operated at the same rate. It is known that
percentages of defective items produced by X, Y, Z are 2,7,12 % respectively. All
the items produced by X, ‏لا‎ ,Z are put into one bin. From this bin ೦೧೮ item 15 drawn
at random and is found to be defective. What is the probability that this item was
produced by Y?

(3)

(2 x 9= 18 marks)