How many distinct 3 digit odd integers are there? Distinct meaning none of the digits repeat.
288
HEY...SORRY...THE ANSWER IS 320
To find the number of distinct 3-digit odd integers where none of the digits repeat, we need to consider a few things:
1. The hundreds digit: Since we cannot have leading zeros in a 3-digit number, the hundreds digit can be any odd digit from 1 to 9 (i.e., 1, 3, 5, 7, or 9).
2. The tens digit: Since none of the digits can repeat, once we have chosen the hundreds digit, there are only 9 remaining digits (0 to 9, excluding the digit chosen for the hundreds place) that can be used in the tens place. However, if the hundreds digit is 9, the tens digit can only be from 0 to 7 because the number has to be odd. If the hundreds digit is any of the other odd numbers (1, 3, 5, or 7), the tens digit can be any odd digit from 0 to 9.
3. The units digit: Again, since none of the digits can repeat, once we have chosen the hundreds and tens digits, there are 8 remaining digits (0 to 9, excluding the digits chosen for the hundreds and tens places) that can be used for the units place. However, the units digit should be an odd digit to make the whole number odd, so there are only 5 odd digits available.
Now, we can calculate the number of distinct 3-digit odd integers by multiplying the choices for each digit:
For the hundreds digit: 5 choices (1, 3, 5, 7, or 9)
For the tens digit:
- If the hundreds digit is 9, there are 8 choices for the tens digit (0 to 7).
- If the hundreds digit is any of the other odd numbers (1, 3, 5, or 7), there are 5 choices for the tens digit (0, 2, 4, 6, or 8).
For the units digit: 5 choices (1, 3, 5, 7, or 9)
Therefore, the total number of distinct 3-digit odd integers is:
(5 choices for the hundreds digit) * (8 choices for the tens digit if the hundreds digit is 9) + (5 choices for the tens digit if the hundreds digit is any other odd number) * (5 choices for the units digit)
= 5 * 8 + 5 * 5
= 40 + 25
= 65
So, there are 65 distinct 3-digit odd integers where none of the digits repeat.