A probability problem by أحمد الحلاق
Probability
Level
3
Find the total number of non-negative integers less than 1200 such that each of its digits are distinct.
The answer is 795.
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1 solution
Break it into cases.
there are 10 1-digit numbers that fit (0 through 9)
For 2-digit numbers, there are 90 to choose from (10-99), but 9 are excluded by duplicating digits, so that's 81 more.
For 3 digit numbers, there are 9 possible first digits (1-9), 9 possible second nonduplicative digits because 0 is now allowed, and 8 possibilities for a 3rd digit. That's 9 * 9 * 8 = 648 more numbers.
For 4 digit numbers, the number must begin with 10, and then there are 8 possible third digits and 7 possible 4th digits, so that's 56 more possibles.
1 0 + 8 1 + 6 4 8 + 5 6 = 7 9 5