Question

a) how many three-digit numbers can be formed from the digits 0,1,2,3,4,5, and 6, if each digit can be used only once?
b) How many of these are odd numbers?
c) How many are greater than 330?

Answer 1

(a) The first digit has 9 options (any numeral 1-9). The second has 9 (any numeral 0-9 minus whichever takes the first position). The third has 8 (any numeral 0-9 not used int he first two digit places).

(b) Take it case by case.

Suppose all digits are odd. Then the first digit has 5 options (1,3,5,7,9). The second has 4 (1,3,5,7,9 except whichever one is the first digit). The third has 3 (1,2,3,5,7,9 not including the first two digit choices).

Suppose the first digit is even. This digit has 4 options (2,4,6,8). The second has 5 (1,3,5,7,9) and the third has 4 (1,3,5,7,9 minus the second digit).

Suppose the first two digits are even. Then the first has 4 options (2,4,6,8) and the second has 4 as well (0 and whichever of 2,4,6,8 isn't the first digit). The third digit has to be odd for the number to be odd, and you have 5 options here (1,3,5,7,9).

Add up all these possible numbers.

(c) Not sure if this question is asking about 3 digit numbers without repeats, like the second question was asking, or if they just want the 3 digit numbers in general that are greater than 330. I'll assume the general number because it's not specified if they want the no-repeat numbers.

The smallest 3 digit number greater than 330 is 331. The largest is 999. What's the difference between 999 and 330?

If the question actually meant to ask for non-repeating digits, try another case-by-case analysis.

If the first digit is 3, then the second digit must be at least 3, and the third digit must be at least 1. That's \(1\times7\times8\) possible numbers of this form. (The first digit must be 3, the second can be any of 3,4,5,6,7,8,9, and the third digit can be any of 1-9 not used up by the second digit.)

If the first digit is greater than 3, then the second digit can take on any number 0-9, and so can the third digit. The first digit has 6 options (4-9), the second has 9 (any of 0-9 except for the first digit) and the third has 8 (any of 0-9 not used in the first or second digits).

Answer 2

thank you :)