The number of ways of arranging 4-digit positive integers with no repeated digits is 4536 ways and number of ways of 4-digit positive integers with repeated digits, but all digits are odd is 625 ways.
In this question,
Positive integers are 0,1,2,3,4,5,6,7,8,9
Total number of integers = 10
This can be solved by permutation concepts.
Case 1: 4-digit positive integers with no repeated digits,
First digit, cannot be zero. So remaining 9 digits.
Second digit, can be any digit other than the first digit. So 9 digits.
Third digit, can be any digits other than first and second. So 8 digits.
Fourth digit, can be any digits other than first, second, third digit. So 7 digits.
Thus, Number of ways of 4-digit positive integers with no repeated digits ⇒ (9)(9)(8)(7)
⇒ 4536 ways.
Case 2: 4-digit positive integers, there may be repeated digits, but all digits are odd
Odd integers are 1,3,5,7,9
Number of digits = 5
In this case, we can repeat the digits. So all places can have 5 possibilities.
Thus number of ways of 4-digit positive integers with repeated digits, but all digits are odd = (5)(5)(5)(5)
⇒ 625 ways.
Hence we can conclude that the number of ways of arranging 4-digit positive integers with no repeated digits is 4536 ways and number of ways of 4-digit positive integers with repeated digits, but all digits are odd is 625 ways.
Learn more about permutation here
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