7/27/2019 Counting & Probability

1/3

1. How many number between 4,000 and 5,000 can be formed with the different digits

2,3,4,5,6,7? Ans:60

2. How many number between 4,000 and 6,000 can be formed with the different digits

2,3,4,5,6,7? Ans: 120

3. How many numbers of three different digits can be formed from integers 3,4,5,6,7?

Ans:60

4. How many numbers of 3 different digits less than 500 can be formed from the integers

1,2,3,4,5,6? Ans:80

5. A four-character password consists of one letter of the alphabet and three different digits

between 0 and 9 inclusive. The letter must appear as the second or third character of the

password. How many different passwords are possible?

(A) 5,040

(B) 18,720

(C) 26,000

(D) 37,440

(E) 52,000

6. A number card numbered from 1 to 30 is drawn randomly. What is the probability of

getting a card having number which is divisible by 5 or 9?

7. If the probability of having rain on any given day in city X is 50%. What is the

probability of having rain in 3 days of a 5 day period?

8. 3 balls are drawn successfully from a bag containing 5 red, 4 blue, and 3 white balls.

Determine the probability that they are all red, if each ball drawn is not replaced?

9. What is the probability of drawing one blue and one red marble from a bag containing 4

red marbles, 5 blue marbles, and 5 green marbles? Marbles are not replaced after being

selected.

10. What is the probability of selecting 2 men from a group of 4 men and 6 women, if the

selection is random?

11. In how many ways can a committee of 3 ladies and 4 gentlemen be appointed from

meeting consisting 15 ladies and 11 gentlemen?

12. A committee is to be chosen from 12 men and 8 women and is consider of 3 men and 2

women. How many different committees can be formed?

13. How many ways are there to arrange the letters in the word elation, if the first and last

letter must each be a vowel?

7/27/2019 Counting & Probability

2/3

14. Let A be the set of primes less than 6, and B be the set of positive odd numbers less than

6. How many different sums of the form a+b are possible if a is in set A and b is in set B?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

15. A drawer contains 2 blue, 4 red, and 2 yellow socks. If 2 socks are to be randomly

selected from the drawer, what is the probability that they will be the same color?

(A) 2/7

(B) 2/5

(C) 3/7

(D)

(E) 3/5

16. Nicole need to form a committee of 3 from a group of 8 research attorneys to study

possible changes to the superior court. If two of the attorneys are too inexperienced to

serve together on the committee, how many different arrangements of committees can

Nicole form? Ans 50

(8 combination 2 6)

17. A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to

select 2 apples from the basket simultaneously at random, what is the probability that the

two apples selected will include the spoiled apple? (Answer: 4/(5combination2) = 4/10

i.e. 2/5)

18. A fair two sided coin is flipped 6 times. What is the probability that tails will be the result

at least twice but not more than 5 times?

(a) 5/8 (b) 3/4 (c) 7/8 (d) 57/64 (e) 15/16

19. From the even numbers between 1 and 9, two different even numbers are to be chosen at

random, what is the probability that their sum will be 8? (26 & 62 i.e. 2/4 combination 2

i.e. 2/12 i.e. 1/6)

(a) 1/6 (b) 3/16 (c) 1/4 (d) 1/3 (e) 1/9

7/27/2019 Counting & Probability

3/3

20. There are four hotels in a town. If three men check into hotel in a day. What is the

probability that each check in a different hotel? (4*3*2)/(4*4*4)

21. Find the probability that in 5 tossing a perfect coin turns head at least three times in a

succession? (3+2+1)/32

22. Twenty books are placed at random in a shelf. Find the probability that a particular pair

of books in the shelf be always together? (Answer: 19/20 Permutation 2)

23. A certain university will select 1 of 7 candidates eligible to fill a position in the

mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the

computer science department. If none of the candidates is eligible for a position in both

departments, how many different sets of three candidates are there to fill the 3 positions?

(Answer: 7 Combination 1 * 10 Combination 2)

24.In how many ways can 4 boys and 3 girls be seated in a row containing

7 seats, if the boys and girls must be seated alternatively? ( Answer:

4*3*3*2*2*1*1)

(a) 140 (b) 141 (c) 142 (d)

143 (e) 144

25. A person has 12 acquaintances of which 8 are relatives. In how many ways can he invite

7 guests so that 5 of them are relatives? (Answer: 8 combination 5 * 4 combination 2)