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1.  
In how many different ways can the letters of the word ATTEMPT be arranged?

Correct Answer: D [840]

Explanation:

7!/3! = 5,040/6 = 840 (T appears three times).

2.  
A committee of 3 men and 2 women is to be chosen from 7 men and 5 women. In how many ways can it be done?

Correct Answer: E [350]

Explanation:

⁷C₃ × ⁵C₂ = 35 × 10 = 350.

3.  
In how many arrangements of the letters of the word PLANET do the two vowels come together?

Correct Answer: B [240]

Explanation:

Glue A, E: 5! × 2! = 240.

4.  
In how many ways can 6 friends sit around a round table?

Correct Answer: A [120]

Explanation:

(6 − 1)! = 120.

5.  
Two cards are drawn together from a pack of 52. The probability that both are spades is:

Correct Answer: C [1/17]

Explanation:

¹³C₂/⁵²C₂ = 78/1,326 = 1/17.

6.  
Two dice are thrown. The probability that the sum is at least 10 is:

Correct Answer: C [1/6]

Explanation:

Sums 10, 11, 12 → 3 + 2 + 1 = 6 ways → 6/36 = 1/6.

7.  
A bag contains 6 red and 4 green balls. Two balls are drawn at random. The probability that one is red and one is green is:

Correct Answer: E [8/15]

Explanation:

⁶C₁ × ⁴C₁/¹⁰C₂ = 24/45 = 8/15.

8.  
A coin is tossed 5 times. The probability of getting at least one tail is:

Correct Answer: A [31/32]

Explanation:

1 − P(no tail) = 1 − 1/32 = 31/32.

9.  
Quantity I: Number of ways to arrange 4 different books on a shelf.
Quantity II: Number of ways to choose 2 students out of 7.

Correct Answer: A [Quantity I > Quantity II]

Explanation:

I: 4! = 24. II: ⁷C₂ = 21. I > II.

10.  
How many handshakes took place at the meeting?
I. Every person shook hands with every other person exactly once.
II. There were 10 people at the meeting.

Correct Answer: D [Both statements together are needed]

Explanation:

Need both: the rule (I) and the number of people (II) → ¹⁰C₂ = 45.

11.  
Trick Spotter: in how many ways can 5 boys sit in a row if two particular boys always sit together? Which is the quickest correct working?

Correct Answer: B [4! × 2!]

Explanation:

Glue the two boys: 4 units → 4!

12.  
Which of the following is/are correct?
I. ⁷C₃ = ⁷C₄
II. 0! = 0
III. ⁵P₃ = 60

Correct Answer: C [Only I and III]

Explanation:

I true (35 = 35); II false (0!

13.  
How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if no digit is repeated?

Correct Answer: B [24]

Explanation:

Last digit 2 or 4 (2 ways), then 4 × 3 for the other places → 24.

14.  
A card is drawn from a pack of 52. The probability that it is a red face card is:

Correct Answer: D [3/26]

Explanation:

Red face cards: J, Q, K of hearts and diamonds = 6 → 6/52 = 3/26.

15.  
Reverse question: ⁿC₂ = 28. What is n?

Correct Answer: C [8]

Explanation:

n(n − 1)/2 = 28 → n(n − 1) = 56 → n = 8.

16.  
A year with 366 days is chosen at random. The probability that it has 53 Mondays is:

Correct Answer: C [2/7]

Explanation:

52 full weeks + 2 extra days; the 2 days include a Monday in 2 of the 7 possible pairs → 2/7.

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