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 Question from camille a student: I need all of the 4 digit combinations that starts with number 3 and digits can't be repeated

Hi Camille,

Suppose you wanted to list all such numbers. Starting from left to right the first digit is a 3. For the next digit you have 9 choices, 0, 1, 2, 4, 5, 6, 7 ,8 or 9. Hence, so far, your list is

30
31
32
34
35
36
37
38
39

Regardless of which 2 digit number you have you can extend it to a 3 digit number in 8 ways. Thus, for example, if you have the 2 digit number 37 you can extend it to

370
371
372
374
375
376
378
379

Since each of the nine, 2 digit numbers can be extended to eight 3 digit numbers your list now has $9 \times 8 = 72,$ 3 digit numbers.

What happens when you add a fourth digit?

Penny

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