VI. Combinatorics Using the digits 0, 1, 2, and e, form four-digit numbers in such a way that the digits in the number

Combinatorics:
Combinatorics is a branch of mathematics that deals with counting, arranging, and organizing objects. It involves various techniques and formulas to solve problems related to permutations and combinations.

Инструкция:
To solve the given problem, we need to use the digits 0, 1, 2, and e to form four-digit numbers without repeating any digit.

16. To determine if 1240 is the smallest of these numbers, we need to consider all possible combinations. Since the numbers cannot repeat, the first digit can only be 1 or 2. If we choose 1 as the first digit, the remaining digits can be arranged in 3! = 6 ways. Similarly, if we choose 2 as the first digit, there are again 6 ways to arrange the remaining digits.

Therefore, there are a total of 6 + 6 = 12 possible four-digit numbers without any repetition. We need to find the smallest among these numbers. After considering all possibilities, we find that the smallest four-digit number is 1204, not 1240. Hence, the statement is false.

17. To determine the number of four-digit numbers greater than 4000, we need to analyze the first digit. Since the first position can only be filled with 2 or e (assuming e represents some other digit), we have 2 possibilities. For the remaining three positions, we have 3! = 6 arrangements.

Thus, there are a total of 2 * 6 = 12 four-digit numbers greater than 4000. The statement is false as it claims that exactly 6 such numbers exist.

18. To calculate the total number of odd four-digit numbers, we know that the last digit must be 1 or e (assuming e represents some other odd digit). We have 2 possibilities for the last digit. For the remaining three positions, we have 3! = 6 arrangements.

Therefore, there are a total of 2 * 6 = 12 odd four-digit numbers.

Демонстрация:
Consider forming four-digit numbers using the digits 0, 1, 2, and e without repetition. Determine if the following statements are true:
16. Is 1240 the smallest four-digit number?
17. How many four-digit numbers are greater than 4000?
18. What is the total number of odd four-digit numbers?

Совет:
When solving combinatorics problems, it is important to carefully analyze the given conditions and consider all possible combinations. Keep track of the number of arrangements and possibilities for each digit or position. Use the factorial function (!) to calculate the number of arrangements.

Дополнительное задание:
Using the digits 0, 1, 2, and 3, form four-digit numbers without repetition. How many four-digit numbers can be formed?