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how many 4-digit numbers can be formed with the 10 digits 0,1,2,3...9 of
(a)repetitions are allowed,
Two ways to get the answer:
1. There are 9999 integers starting with 1 and ending with 9999. But the
999 integers starting with 1 and ending with 999 have less than 4 digits,
so the desired number is 9999-999 or 9000 ways.
2. There are 9 ways to pick the first digit (1 through 9), there are 10
ways to pick the 2nd, 3rd, and 4th digits, so that 9×10×10×10 = 9000 ways.
There are 9 ways to pick the first digit (it can't be 0), 9 ways to pick the
2nd digit (it can be 0, just not what we picked for the 1st digit), 8 ways to
pick the third digit, and 7 ways to pick the fourth digit. That's 9×9×8×7,
or 4536 ways.
(c) the last digit must be 0 and repetitions are not allowed:
We can choose the first digit any of 9 ways, the second digit any of 8 ways,
the third digit any of 7 ways and the last digit only 1 way (a 0).
That's 9×8×7×1 = 504 ways.