Question

We construct 4-digit numbers, which are chosen from any of the following seven digits: 0,1,3,5,7,8, 9. Answer the following: a) How many different numbers are possible if a digit can be used more than once? b) How many different numbers are there that contain no identical digits? c) What is the probability of a number having no digit ' 7 ' and containing two or more identical digits?

Answer 1

a) The number of different numbers possible if a digit can be used more than once is 2401. b) The number of different numbers that contain no identical digits is 840. c) The probability of a number having no digit '7' and containing two or more identical digits is approximately 0.458.

Step-by-step explanation:

a) To find the number of different numbers possible if a digit can be used more than once, we need to identify the number of choices for each digit. Since we have 7 digits to choose from, there are 7 choices for each digit. Since we are constructing a 4-digit number, the total number of possibilities is calculated as 7^4 = 2401.

b) To find the number of different numbers that contain no identical digits, we can start by choosing any digit from the 7 available choices for the first digit. For the second digit, there are 6 choices remaining (since we can't use the same digit as the first digit). Similarly, for the third digit, there are 5 choices remaining, and for the fourth digit, there are 4 choices remaining. Thus, the total number of possibilities is calculated as 7 * 6 * 5 * 4 = 840.

c) To find the probability of a number having no digit '7' and containing two or more identical digits, we can start by calculating the number of possible numbers without a '7'. From the 7 available digits, we have 6 choices for each digit. Since we are constructing a 4-digit number, the total number of possibilities without a '7' is calculated as 6^4 = 1296. Now, to find the probability of having two or more identical digits, we can calculate the number of possibilities without any restrictions (7^4 = 2401) and subtract the number of possibilities without two or more identical digits (6^4 = 1296). The probability is then calculated as (2401 - 1296) / 2401 ≈ 0.458.

Answer 2

a) There are 2401 different numbers possible if a digit can be used more than once. b) There are 840 different numbers that contain no identical digits. c) The probability of a number having no digit '7' and containing two or more identical digits is 7/6.

Step-by-step explanation:

a) To find the number of different numbers possible with 4 digits, we need to consider that each digit can be chosen from 7 options (0, 1, 3, 5, 7, 8, 9) and can be used more than once. So, for each digit, there are 7 choices, and since there are 4 digits, the total number of different numbers is 7^4 = 2401.

b) To find the number of different numbers that contain no identical digits, we need to consider that each digit can be chosen from 7 options (0, 1, 3, 5, 7, 8, 9) and can only be used once. So, for the first digit, there are 7 choices, for the second digit, there are 6 choices (as one digit has already been used), for the third digit, there are 5 choices, and for the fourth digit, there are 4 choices. Therefore, the total number of different numbers is 7 * 6 * 5 * 4 = 840.

c) To find the probability of a number having no digit '7' and containing two or more identical digits, we first need to find the total number of different numbers possible without the digit '7'. Using the same reasoning as in part (b), the number of different numbers without '7' is 6 * 6 * 5 * 4 = 720. Then, we need to find the number of different numbers with two or more identical digits, which is 840 (as calculated in part (b)). Therefore, the probability is 840/720 = 7/6.