These 2 questions have to do with permutations
answer to #1-26
answer to #2-3439
1. In how many ways can you form a 3 digit number, using only the digits of the number 21 150?
2. How many natural numbers less than 10 000 include at least one nine.
please explain to me how you get the anwers
Sure! I'll explain how to get the answers to both questions.
1. To find the number of ways you can form a 3-digit number using the digits 2, 1, 1, 5, and 0, you can use the concept of permutations.
First, we need to determine the number of choices for each digit's position. Since the question specifies that we can only use the digits from the number 21,150, we have the following possibilities:
For the hundreds place digit, we have two options: 2 and 1.
For the tens place digit, we have four options: 2, 1, 1, and 5.
For the ones place digit, we have three options: 0, 1, and 5.
To find the total number of ways, we multiply the number of options for each position: 2 x 4 x 3 = 24.
Therefore, there are 24 different three-digit numbers that can be formed using the digits 2, 1, 1, 5, and 0.
2. To find the number of natural numbers less than 10,000 that include at least one nine, we can subtract the count of numbers that do not include the digit nine from the total count of numbers.
The total count of natural numbers less than 10,000 is 10,000 - 1 = 9,999 (excluding zero).
To count the numbers that do not include the digit nine, we need to consider each digit's position.
For the thousands place digit, we have ten possible digits (0-9), but we exclude the digit nine. So, there are nine options.
For the hundreds, tens, and ones place digits, we have ten possibilities each because we can use any digit from 0 to 9.
To find the count of numbers that do not include the digit nine, we multiply the number of options for each position: 9 x 10 x 10 x 10 = 9,000.
To find the count of numbers that include at least one nine, we subtract the count of numbers that do not include the digit nine from the total count: 9,999 - 9,000 = 999.
Therefore, there are 999 natural numbers less than 10,000 that include at least one nine.