Question

how many rising 4-digit numbers can you make from decimal digits? a rising number is a positive integer where each digit is larger than the one to its left.

Answer 1

To create a rising 4-digit number, we start with 10 choices for the first digit and reduce the number of choices for each subsequent digit. The total number of rising 4-digit numbers that can be made from decimal digits is 5,040.

Step-by-step explanation:

To create a rising 4-digit number, we need to choose a larger digit for each position to the right. Since there are 10 digits (0-9) to choose from, the first digit can be any of the 10 digits. For the second digit, we can only choose from 1-9 because a rising number cannot start with 0. Similarly, for the third digit, we can choose from 2-9, and for the fourth digit, we can choose from 3-9.

Therefore, the number of rising 4-digit numbers that can be made from decimal digits is:

10 choices for the first digit × 9 choices for the second digit × 8 choices for the third digit × 7 choices for the fourth digit = 5,040.

Answer 2

There are 126 rising 4-digit numbers that can be created from decimal digits, calculated using the combinatorial formula C(9, 4), which represents choosing 4 distinct digits from the 9 available.

Step-by-step explanation:

The question asks how many rising 4-digit numbers can be made from the decimal digits. A rising number is defined as a positive integer where every digit is larger than the one to its left. An important thing to note here is that each digit can be used only once because it must be smaller than the one to its right.

Since there are ten digits (0-9) to choose from, and we are looking at 4-digit numbers, the first digit can be any digit from 1 to 9 (0 can't be the first digit as it would not make the number a 4-digit number). The remaining digits must be selected from the remaining pool of digits such that each subsequent digit is larger than the previous one.

To calculate this, we can simply choose 4 distinct digits from the 9 available (since the first digit cannot be 0). The number of ways to choose 4 digits out of 9, without regards to order, is given by the combinatorial formula for combinations (since the order of the digits matters for our problem, they will be arranged in ascending order by default after choosing them). The formula for combinations is expressed as C(n, k) = n! / (k! * (n-k)!), where n is the total number of items to pick from, k is the number of items to pick, and ! denotes factorial.

Therefore, for our case, the combination would be C(9, 4) = 9! / (4! * (9-4)!) = 9! / (4! * 5!) = 126. Thus, there are 126 rising 4-digit numbers that can be made from decimal digits.