1. The street of a city are arranged like the lines of a
chessboard , there are m street running north and south and n east
and west . Find the no. ways in which a man can travel from N.W to
the S.E corner, going by the shortest possible distance.

2. How many different arrangement can be made out of the letters in
the ex-pression a^3b^2c^4 when written at full length?

Ans. 1260.

3. How many 7-digit numbers exist which are divisible by 9 and whose
last but one digit is 5?

4. You continue flipping a coin until the number of heads equals the
number of tails. I then award you prize money equal to the number of
flips you conducted .How much are you willing to pay me to play this
game?

5. Consider those points in 3-space whose three coordinates are all
nonnegative integers, not greater than n. Determine the number of
straight lines that pass through n of these points.

6 Four figures are to be inserted into a six-page essay, in a given
order. One page may contain at most two figures. How many different
ways are there to assign page numbers to the figures under these
restrictions?

7. How many (unordered) pairs can be formed from positive integers
such that, in each pair, the two numbers are coprime and add up to
285?

8. ( 1+ x) ^ n – nx is divisible by

A. x B. x.x C. x.x.x. D. None of these

9. The root of the equation (3-x)^4 + (2-x)^4 = (5-2x)^4 are

a. ALL REAL B. all imaginary C. two real and 2 imaginary D. None of
these

10. The greatest integer less than or equal to ( root(2)+1)^6 is

A. 196 B.197 C.198 D.199.

11. How many hundred-digit natural numbers can be formed such that
only even digits are used and any two consecutive digits differ by 2?

12. If x1 , x2, x3 are the root of x^3-1=0 , then

A. x1+x2+x3 not equal to 0 B. x1.x2.x3 not equal to 1 C.(x1+x2+x3)^2
= 0 D. None of these

13. How many different ways are there to arrange the numbers
1,2,...,n in a single row such that every number, except the number
which occurs first, is preceded by at least one of its original
neighbours?

14. There are N people in one room. How big does N have to be until
the probability that at least two people in the room have the same
birthday is greater than 50 percent? (Same birthday means same month
and day, but not necessarily same year.)

15. A 25 meter long wound cable is cut into 2 and 3 meter long
pieces. How many different ways can this be done if also the order of
pieces of different lengths is taken into account?

16. You are presented with a ladder. At each stage, you may choose to
advance either one rung or two rungs. How many different paths are
there to climb to any particular rung; i.e. how many unique ways can
you climb to rung "n"? After you've solved that,
generalize. At each stage, you can advance any number of rungs from 1
to K. How many ways are there to climb to rung "n"?

17. Find the no. of rational number m/n where m ,n are relatively
prime positive no. satisfying m<n and mn = 25!..?

18. The master of a college and his wife has decided to throw a party
and invited N guest and their spouses. On the night of the party, all
guests turned up with their spouse, and they all had a great time.
When the party was concluding, the master requested all his guests
(including his wife, but not himself) to write down the number of
persons they shook hands with, and to put the numbers in a box. When
the box was opened, he was surprised to find all integers from 0 to
2N inclusive. Assuming that a person never shake hands with their own
spouse and that no one lied, how many hands did the master shake?

19. A rectangle OACB with 2 axes as sides, the origin O as a vertex
is drawn in which length OA is 4 time the width OB . A circle is
drawn passing through the point B and C touching OA at its min-point,
thus dividing the rectangle into 3 parts. Find the ratio of areas of
these 3 parts. ?

20. A rod is broken into 3 parts ; the 2 break point are chosen at
random.Find the probability that the 3 parts can be joined at the
ends to form a triangle.?

21. The distance between town A and B is 1000 k.m. A load containing
10000 kg of jaggery is to be transferred from A to B. A camel is the
only source of transport. However, the camel is unique. It can
maximum carry a load of 1000 kgs. Further, for it is a voracious
eater, and for every k.m. it travels, it eats a kilo of the jaggery
which it is carrying. What is the maximum amount of jaggery that can
be transferred from A to B ?

22. The ticket office at a train station sells tickets to 200
destinations. One day, 3800 passengers buy tickets. Then minimum no
of destinations receive the same number of passengers is

A.6 B.5 C.7 D.9

23. A hexagon with sides of length 2, 7, 2, 11, 7, 11 is inscribed in
a circle. Find the radius of the circle.

24.On the sides of an acute triangle ABC are constructed externally a
square, a regular n-gon and a regular m-gon (m, n > 5) whose
centers form an equilateral triangle. Prove that m = n = 6, and find
the angles of ABC.

25. The area of triangle formed by the point (p,2-2p) , (1-p,2p) and
(-4-p,6-2p) is 70 units. How many integral values of p are possible.?

A. 2 B. 3 C. 4 D. None of these

26. In a park, 10000 trees have been placed in a square lattice.
Determine the maximum number of trees that can be cut down so that
from any stump, you cannot see any other stump. (Assume the trees
have negligible radius compared to the distance between adjacent
trees.)

A. 2500 B.4900 C.6400 D. none of these.

27. A certain city has a circular wall around it, and this wall has
four gates pointing north, south, east and west. A house stands
outside the city, three miles north of the north gate, and it can
just be seen from a point nine miles east of the south gate. The
problem is to find the diameter of the wall that surrounds the city.

28. Income of A,B and C are in the ratio 7:10:12 and their expenses
in the ratio 8:10:15 . If A saves 20% of his income then B’s saving
is what % more/less than that of C’s

A.100% more B. 120% more C. 80% less D. 40% less

29. A triangle ABC has positive integer sides, angle A = 2 angle B
and angle C > 90. Find the minimum length of the perimeter of ABC.

30. We have k identical mugs. In an n-storey building, we have to
determine the highest floor from which, when a mug is dropped, it
still does not break. The experiment we are allowed to do is to drop
a mug from a floor of our choice. How many experiments are necessary
to solve the problem in any case, for sure? [/glow][/color][/font]

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