6
Question
The number of four digit numbers that can be formed from the digit 0,1,2,3,4,5 with at least one digit repeated is ?
The number of four digit numbers that can be formed from the digit 0,1,2,3,4,5 with at least one digit repeated is ?
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Solution
The correct option is D 750
Since we are looking
for 44-digit numbers, the first (thousands) digit can't be 00.
Let's split cases:
Case 1: first digit is 11.In this case, we pick 33 digits
in order out of the remaining 77, there are 7⋅6⋅5=2107⋅6⋅5=210 possibilities.
Case 2:
first digit is not 11.
There are 66 ways
to choose the first digit (2,3,...,72,3,...,7), 33 ways to place the
digit 11 (hundreds, tens, ones), and 6⋅56⋅5 (pick 22 in
order from the remaining 66 digits) ways to choose the other digits.
So there are 6⋅3⋅6⋅5=5406⋅3⋅6⋅5=540 possibilities in this case.
Total In total, there
are 210+540=750210+540=750 possibilities to choose the number.
Since we are looking for 44-digit numbers, the first (thousands) digit can't be 00.
Let's split cases:
Case 1: first digit is 11.In this case, we pick 33 digits in order out of the remaining 77, there are 7⋅6⋅5=2107⋅6⋅5=210 possibilities.
Case 2: first digit is not 11.
There are 66 ways to choose the first digit (2,3,...,72,3,...,7), 33 ways to place the digit 11 (hundreds, tens, ones), and 6⋅56⋅5 (pick 22 in order from the remaining 66 digits) ways to choose the other digits. So there are 6⋅3⋅6⋅5=5406⋅3⋅6⋅5=540 possibilities in this case.
Total In total, there are 210+540=750210+540=750 possibilities to choose the number.
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