Permutations

Step 1: Standard derangement formula.
The number of ways to permute 5 items such that none is in its original place (a “derangement”) is denoted by . A known formula: Hence So there are 44 derangements for 5 items.

Step 2: Probability that no man is matched with his wife.
Total ways to match 5 men's names with 5 wives' names is . The favorable ways (derangements) is 44. Thus

We are given the digits and need to form 4-digit numbers between 6000 and 9999 without repetition.

Step 1: Identify Valid Leading Digits
- A number must be , so the first digit must be 6, 7, or 8 (valid leading digits). - The remaining 3 digits are selected from the remaining 4 digits.

Step 2: Count Even and Odd Numbers
- A number is even if its last digit is even ( ). - A number is odd if its last digit is odd ( ).

Step 3: Count Odd and Even Cases
1. Case 1: Last digit is even (6 or 8) - First digit choices: (3 choices). - Last digit choices: or (2 choices). - Middle two digits are chosen from remaining 3 digits: . 2. Case 2: Last digit is odd (3,5,7) - First digit choices: (3 choices). - Last digit choices: (3 choices). - Middle two digits are chosen from remaining 3 digits: .

Step 4: Compute Difference Between Odd and Even Numbers

Step 5: Conclusion
Thus, the difference between the number of odd and even numbers is:

Step 1: Understanding the Given Equation
The equation given is a cubic equation: We analyze different conditions based on the values of :
- All roots are negative: This happens when , which corresponds to IV.
- Two roots are complex: This occurs in the range , corresponding to II.
- Two roots are positive: This occurs when , which corresponds to V.
- All roots are real and distinct: This happens for , which corresponds to III.
Step 2: Evaluating the Given Options
- Option (1): Incorrect, as incorrect matches are present.
- Option (2): Incorrect, as incorrect matches are present.
- Option (3): Incorrect, as incorrect matches are present.
- Option (4): Correct, as all matches are accurate.
Thus, the correct answer is

Option (4).

Step 1: Understanding the Given Relatives
The man's relatives consist of: - 4 ladies - 3 gents The wife's relatives consist of: - 3 ladies - 4 gents The total relatives are 14, but we need to select exactly 3 ladies and 3 gents ensuring that 3 are from the man's side and 3 from the wife's side.

Step 2: Choosing 3 Ladies
Since 3 ladies must be selected from both families: - Choose 2 ladies from the 4 ladies among the man's relatives: - Choose 1 lady from the 3 ladies among the wife's relatives: Total ways to select ladies:

Step 3: Choosing 3 Gents
Since 3 gents must be selected from both families: - Choose 1 gent from the 3 gents among the man's relatives: - Choose 2 gents from the 4 gents among the wife's relatives: Total ways to select gents:

Step 4: Compute the Total Number of Ways
Thus, the number of ways to invite 3 ladies and 3 gents while ensuring an equal split from both families is:

Step 1: Identify Consonants and Vowels
The word "COMBINATIONS" consists of 11 letters: - Vowels: (5 vowels) - Consonants: (7 consonants) To ensure that no two vowels are adjacent, we first arrange the consonants in a circular arrangement.

Step 2: Arrange Consonants in a Circle
Since circular permutations of distinct objects is given by , the consonants can be arranged in:

Step 3: Placing Vowels in Gaps
Once the consonants are placed in a circle, they create 7 gaps. The 5 vowels must be placed in these gaps. The number of ways to choose 5 gaps from 7 is: Since vowels include repetitions, we divide by factorials of repeated letters:

Final Answer: