Question

Four digit numbers with all digits distinct are formed using the digits 1, 2, 3, 4, 5, 6, 7 in all possible ways. If \( p \) is the total number of numbers thus formed and \( q \) is the number of numbers greater than 3400 among them, then \( p : q = \):

• A. \( 3:2 \)
• B. \( 4:3 \)
• C. \( 6:5 \)
• D. \( 7:4 \)

Hint:

When calculating permutations with restrictions, consider the number of possibilities for each digit step-by-step and account for the restrictions as you go.

Answer 1

The Correct Option is A

We are asked to form four-digit numbers using the digits \( 1, 2, 3, 4, 5, 6, 7 \) with all digits distinct. Step 1: Calculating \( p \), the total number of four-digit numbers The number of ways to choose the first digit is 7 (since it can be any digit from 1 to 7). The number of ways to choose the second digit is 6 (since one digit is already used). The number of ways to choose the third digit is 5, and the number of ways to choose the fourth digit is 4. Thus, the total number of numbers is: \[ p = 7 \times 6 \times 5 \times 4 = 840. \] Step 2: Calculating \( q \), the number of numbers greater than 3400 For numbers greater than 3400, the first digit must be 3, 4, 5, 6, or 7. - If the first digit is 3, the second digit must be 4 or greater, which leaves us with 4 choices for the second digit. The remaining 2 digits can be chosen in \( 5 \times 4 = 20 \) ways. Thus, the number of such numbers is \( 1 \times 4 \times 5 \times 4 = 80 \). - If the first digit is 4, the second digit can be any of the remaining digits (6 choices), and the third and fourth digits can be chosen in \( 5 \times 4 = 20 \) ways. Thus, the number of such numbers is \( 1 \times 6 \times 5 \times 4 = 120 \). - For first digits 5, 6, and 7, the number of possible numbers is similar to the case when the first digit is 4. Thus, the total number of numbers greater than 3400 is: \[ q = 80 + 120 + 120 + 120 + 120 = 560. \] Step 3: Finding the ratio \( p : q \) The ratio is: \[ p : q = 840 : 560 = 3 : 2. \] Thus, the ratio \( p : q \) is \( 3:2 \).

Answer 2

Problem: Four-digit numbers with all digits distinct are formed using the digits 1, 2, 3, 4, 5, 6, 7 in all possible ways. Let \( p \) be the total number of such numbers formed. Let \( q \) be the number of these numbers that are greater than 3400. Find the ratio \( p : q \).

Step 1: Calculate total numbers \( p \) - The first digit can be any of the 7 digits (1 to 7), so 7 choices. - The second digit can be any of the remaining 6 digits. - The third digit can be any of the remaining 5 digits. - The fourth digit can be any of the remaining 4 digits. Thus, \[ p = 7 \times 6 \times 5 \times 4 = 840. \]

Step 2: Calculate numbers greater than 3400, \( q \) The number is greater than 3400 if: - The first digit is greater than 3, or - The first digit is 3 and the number formed by the last three digits is greater than 400.
Case 1: First digit > 3 Possible digits for the first digit: 4, 5, 6, 7 → 4 choices. Remaining three digits are chosen from the remaining 6 digits without repetition. Number of ways: \[ 4 \times 6 \times 5 \times 4 = 480. \]
Case 2: First digit = 3 and number > 3400 The last three digits form a number greater than 400, meaning the second digit (hundreds place) is at least 4. - Second digit choices: 4, 5, 6, 7 → 4 choices. - Third digit: from remaining digits excluding the first two → 5 choices. - Fourth digit: from remaining digits excluding the first three → 4 choices. Number of ways: \[ 1 \times 4 \times 5 \times 4 = 80. \]

Total numbers greater than 3400: \[ q = 480 + 80 = 560. \]

Step 3: Calculate the ratio \( p : q \) \[ p : q = 840 : 560 = 3 : 2. \]

Final answer: \[ \boxed{3 : 2}. \]