1. **Problem Statement:** Calculate the number of possible telephone numbers, dressing combinations, security alarm codes, and four-digit numbers based on given constraints. 2. **Telephone Numbers (Question 3):** - i. Area codes: The first digit cannot be zero, so it can be any digit from 1 to 9 (9 options). The next two digits can be any digit from 0 to 9 (10 options each). Total area codes = $9 \times 10 \times 10 = 900$. - ii. Local numbers: The first digit cannot be zero (9 options), and the remaining six digits can be any digit (10 options each). Total local numbers = $9 \times 10^6 = 9,000,000$. - iii. Total telephone numbers = area codes $\times$ local numbers = $900 \times 9,000,000 = 8,100,000,000$. 3. **Dressing Combinations (Question 4):** - Number of suits = 4 - Number of ties = 3 - Number of shoes = 2 - Total ways = $4 \times 3 \times 2 = 24$. 4. **Security Alarm Codes (Question 5):** - i. No digit repeated: For a 4-digit code, digits 0-9 (10 digits). Number of codes = permutations of 10 digits taken 4 at a time = $P(10,4) = 10 \times 9 \times 8 \times 7 = 5040$. - ii. Digits may repeat: Each digit can be any of 10 digits. Number of codes = $10^4 = 10,000$. 5. **Four-digit numbers from digits 0,1,2,3,5,6 with no repetition (Question 6):** - Digits available = 6 - First digit cannot be zero (to form a valid 4-digit number), so first digit options = 5 (1,2,3,5,6) - Remaining 3 digits chosen from remaining 5 digits without repetition. - Number of such numbers = $5 \times P(5,3) = 5 \times (5 \times 4 \times 3) = 5 \times 60 = 300$. **Final answers:** - Question 3: i. 900, ii. 9,000,000, iii. 8,100,000,000 - Question 4: 24 - Question 5: i. 5040, ii. 10,000 - Question 6: 300