In a game of 100 points, A can give B 20 points and C 28 points. Then, B can give C:

Ratio of the speeds of A and B =	5	: 1 = 5 : 3.
	3

Thus, in race of 5 m, A gains 2 m over B.

2 m are gained by A in a race of 5 m.

80 m will be gained by A in race of	❨	5	x 80	❩m	= 200 m.
		2

∴ Winning post is 200 m away from the starting point.

A : B = 100 : 90.

A : C = 100 : 87.

B	=	B	x	A	=	90	x	100	=	30	.
C		A		C		100		87		29

When B runs 30 m, C runs 29 m.

When B runs 180 m, C runs	❨	29	x 180	❩m	= 174 m.
		30

∴ B beats C by (180 - 174) m = 6 m.

A : B = 60 : 45.

A : C = 60 : 40.

∴

B

=

❨

B

x

A

❩

=

❨

45

x

60

❩

=

45

=

90

= 90 : 80.

C

A

C

60

40

40

80

∴ B can give C 10 points in a game of 90.

A's speed =	❨	5 x	5	❩m/sec	=	25	m/sec.
			18			18

Time taken by A to cover 100 m =	❨	100 x	18	❩sec	= 72 sec.
			25

∴ Time taken by B to cover 92 m = (72 + 8) = 80 sec.

∴ B's speed =	❨	92	x	18	❩kmph	= 4.14 kmph.
		80		5

To reach the winning post A will have to cover a distance of (500 - 140)m, i.e., 360 m.

While A covers 3 m, B covers 4 m.

While A covers 360 m, B covers	❨	4	x 360	❩m	= 480 m.
		3

Thus, when A reaches the winning post, B covers 480 m and therefore remains 20 m behind.

∴ A wins by 20 m.

Distance covered by B in 9 sec. =	❨	100	x 9	❩m	= 20 m.
		45

∴ A beats B by 20 metres.

Let S be the sample space.

Then, n(S)	= number of ways of drawing 3 balls out of 15
	= 15C3
	= (15 x 14 x 13) (3 x 2 x 1)
	= 455.

Let E = event of getting all the 3 red balls.

∴ n(E) = 5C3 = 5C2 =	(5 x 4)	= 10.
	(2 x 1)

∴ P(E) =	n(E)	=	10	=	2	.
	n(S)		455		91

Here, S = {1, 2, 3, 4, ...., 19, 20}.

Let E = event of getting a multiple of 3 or 5 = {3, 6 , 9, 12, 15, 18, 5, 10, 20}.

∴ P(E) =	n(E)	=	9	.
	n(S)		20

Clearly, n(S) = (6 x 6) = 36.

Let E = Event that the sum is a prime number.

Then E

= { (1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3),       (5, 2), (5, 6), (6, 1), (6, 5) }

∴ n(E) = 15.

∴ P(E) =	n(E)	=	15	=	5	.
	n(S)		36		12

Clearly, there are 52 cards, out of which there are 12 face cards.

∴ P (getting a face card) =	12	=	3	.
	52		13