**1.How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?**

**2.Two dice are tossed. The probability that the total score is a prime number is:**

**3.There are 6 boxes numbered 1, 2,….6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is**

**4.In how many ways can the letters of the word “PROBLEM” be rearranged to make 7 letter words such that none of the letters repeat?**

**5.A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his target?**

**6.Rohit has 9 pairs of dark Blue socks and 9 pairs of Black socks. He keeps them all in a same bag. If he picks out three socks at random what is the probability he will get a matching pair?**

**7.There are 5 Rock songs, 6 Carnatic songs and 3 Indi pop songs. How many different albums can be formed using the above repertoire if the albums should contain at least 1 Rock song and 1 Carnatic song?**

**8.From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?**

**9.A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?**

**10. In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?**

**11.A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?**

**12.In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?**

**13.Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?**

**14.In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?**

**15.From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?**

**Answers:-**

**1.D**

Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

Required number of numbers = (1 x 5 x 4) = 20.

**2.B**

**3.B**

If only one of the boxes has a green ball, it can be any of the 6 boxes. So, this can be achieved in 6 ways.

If two of the boxes have green balls and then there are 5 consecutive sets of 2 boxes. 12, 23, 34, 45, 56.

Similarly, if 3 of the boxes have green balls, there will be 4 options.

If 4 boxes have green balls, there will be 3 options.

If 5 boxes have green balls, then there will be 2 options.

If all 6 boxes have green balls, then there will be just 1 options.

Total number of options = 6 + 5 + 4 + 3 + 2 + 1 = 21.

**4.A**

There are seven positions to be filled.

The first position can be filled using any of the 7 letters contained in PROBLEM.

The second position can be filled by the remaining 6 letters as the letters should not repeat.

The third position can be filled by the remaining 5 letters only and so on.

758

Therefore, the total number of ways of rearranging the 7 letter word = 7*6*5*4*3*2*1 = 7! Ways.

**5.D**

The probability that he will not hit the target in one shot = 1 – 1/4 = 3/4

Therefore, the probability that he will not hit the target in all the four shots =81/256

Hence, the probability that he will hit the target at least in one of the four shots = 1 – 81/256

= 175/256 .

**6.C**

If he picks any of the three socks invariably any two of them should match. Hence the probability is 1.

**7.A**

**8.D**

We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).

Required number of ways = (7C3 x 6C2) + (7C4 x 6C1) + (7C5)

=756

**9.C**

We may have(1 black and 2 non-black) or (2 black and 1 non-black) or (3 black).

Required number of ways = (3C1 x 6C2) + (3C2 x 6C1) + (3C3)

=64

**10.A**

Required number of ways = (7C5 x 3C2) = (7C2 x 3C1)

=63

**11.A**

Total number of balls = (2 + 3 + 2) = 7.

Let S be the sample space.

Then, n(S)= Number of ways of drawing 2 balls out of 7

= 7C2 `

=(7 x 6)/(2 x 1)

= 21.

Let E = Event of drawing 2 balls, none of which is blue.

n(E)= Number of ways of drawing 2 balls out of (2 + 3) balls.

= 5C2

=(5 x 4)/(2 x 1)

= 10.

P(E) =n(E)/n(S=10/.21

**12.A**

Total number of balls = (8 + 7 + 6) = 21.

Let E= event that the ball drawn is neither red nor green

= event that the ball drawn is blue.

n(E) = 7.

P(E) =n(E)/n(S)=7/21=1/3.

**13.B**

In a simultaneous throw of two dice, we have n(S) = (6 x 6) = 36.

Then, E = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4),

(3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1),

(6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

n(E) = 27.

P(E) =n(E)/n(S)=27/36=3/4

**14.C**

P (getting a prize) = 10/(10 + 25)=10/35 =2/7 .

**15.D**