Combinations How many 4 digit numbers less than 4000 can be made using.. How many 4 digit numbers less than 4000 can be made using the digits 
1, 2, 3, 5, 7 and 9 if repetition is not permitted?
 A: In order to be less than 4000 you can only use 1, 2, or 3 as the first digit -- so there are 3 ways to choose the first digit.
This leaves you with five digits to choose from for the second digit (you started with six digits to choose from and repetition is not allowed). For the third digit there are four digits to choose from (again, no repetitions), and for the fourth digit there are three digits to choose from.
So overall there are $3 \times 5 \times 4 \times 3 = 180$ numbers.
A: Thre are $3$ choices for the first digit. For each of these, there are $5$ choices for the second digit. For every choice of first and second digit, there are $4$ choices for the third digit. and for every choice of the first three digits, there are $3$ choices for the fourth digit. Thus the total number is $(3)(5)(4)(3)$.
We can save a little time by observing that for every choice of first digit, there are $(5)(4)(3)$ ways to complete the job.
A: $(3)(5)(4)(3)+(6)(5)(4)+(6)(5)+(6)$ you need to account for $3$ digit, $2$ digit and $1$ digit numbers that can be made as they are separate, I made this mistake the first time.
