1. How many positive integers between 1 and 1000, (including both 1 and 1000) are not divisible by 3, 5, and 7?

• 193

• 228

• 475

• 457

Correct Answer : 457

2. How many onto functions are possible from a set of 9 elements to a set of 4 elements?

• 186488

• 180336

• 186480

• 184789

Correct Answer : 186480

3. In how many ways can integers from 1 to 10 be arranged such that no prime number is at its natural place?

4. In a 3 × 3 chessboard:

I) One rook can be placed in 9 ways.

II) Two rooks can be placed in (9 2) ways such that they do not kill each other (non–taking rooks).

III) Three rooks can be placed in 6 ways such that they do not kill each other (non–taking rooks).

Which of the following statements are true?

• I and II

• I and III

• I II and III

• II and III

Correct Answer : I and III

5. How many integer solutions are there for the equation 𝑥+𝑦+𝑧=18, where, 𝑥<5,𝑦<9,𝑧<7 ?

• 1

• 2

• 3

• 4

Correct Answer : 1

6. |𝐴|=10,|𝐵|=20,|𝐶|=30,|𝐴∩𝐵|=4,|𝐵∩𝐶|=5,|𝐴∩𝐶|=6,  |𝐴∩𝐵∩𝐶|=7, then |𝐴∪𝐵∪𝐶| is

• 38

• 52

• 53

• 60

Correct Answer : 52

7. A basketball team has 7 players, each having distinct names. Every player has a jersey with their names on the back of it. If the jerseys are lying in the changing room, turned inside-out. In how many ways can the players wear them so that no one wears the jersey of their name (each one wears someone else's jersey)?

• 3186

• 2049

• 1854

• 5040

Correct Answer : 1854

8. In a class of 120 students, 67 students like Mathematics and 86 students like Physics, while 53 students like both the subjects. What is the total number of students who like neither of the two subjects?

• 67

• 20

• 100

• 49

Correct Answer : 20

9. In how many ways can 5 tigers, 5 lions, and 5 elephants march–past, in the same row, such that no consecutive 5 animals of the same family are together?

10. For some Greek alphabets 𝛼, 𝛽, 𝛾, 𝛿,… (𝑛 − 1)th alphabet, (𝑛)th alphabet, there are 491310 derangements, where 𝛼, 𝛽, 𝛾, 𝛿, 𝜀, 𝜁 appear in the first 6 positions. What is the value of 𝑛 ?

• 11

• 24

• 20

• 13

Correct Answer : 13

WEEK-10 DETAILED SOLUTION