A 5 digit positive integer is a "mountain number" if the first 3 digits are in ascending order and the last 3 digits are in descending order. For example, 35 761 is a mountain number, but 32 323 and 35 655 are not. How many 5 digit numbers greater than 70 000 are mountain numbers?
To find the number of 5-digit mountain numbers greater than 70,000, we can break down the problem into two parts: the ascending part and the descending part.
First, let's consider the ascending part. The first digit can be any number from 8 to 9 because it is greater than 7. For the second digit, it can be any number from the previous digit to 9. Similarly, for the third digit, it can be any number from the previous digit to 9.
For the descending part, we need to consider the last three digits. The last digit can be any number from 0 to 6 because it is less than 7. For the fourth digit, it can be any number from 0 to the previous digit. Similarly, for the fifth digit, it can be any number from 0 to the previous digit.
Now we can calculate the total number of mountain numbers. By multiplying the number of choices for each position, we get:
(2 choices for the first digit) * (2 choices for the second digit) * (2 choices for the third digit) * (7 choices for the fourth digit) * (7 choices for the fifth digit) = 2 * 2 * 2 * 7 * 7 = 196.
Therefore, there are 196 five-digit numbers greater than 70,000 that are mountain numbers.