Fundamental Counting Principle

Question

How many four-digit numbers can be formed from the set $\{ 0, 1, 2, 3,\ldots , 10 \}$ if zero cannot be the first digit and the given conditions are to be satisfied

• Repetitions are allowed and the number must be even.
• Repetitions are allowed and the number must be divisible by $5$.
• The number must be odd and less than $4000$ with repetition allowed.

a. my solution is 2*10*10*10= 2000 because 2 is a even number and there are 10 numbers excluding 0 in set {0,1,2,3..10} and it is 4 digits that's why 2*10*10*10 b.same in letter A but 2 is changed into 5 because it must be divisible by 5 so it is 5*10*10*10=5000 c.same to A and B... Only I changed it to 3 so my solution is 3*10*10*10 = 3000

Answer 1

$9\cdot10\cdot10\cdot5=4500$:

• Digit #$1$ can be any of the $ 9$ digits in $[ 1,2,3,4,5,6,7,8,9]$
• Digit #$2$ can be any of the $10$ digits in $[0,1,2,3,4,5,6,7,8,9]$
• Digit #$3$ can be any of the $10$ digits in $[0,1,2,3,4,5,6,7,8,9]$
• Digit #$4$ can be any of the $ 5$ digits in $[0, 2, 4, 6, 8 ]$

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$9\cdot10\cdot10\cdot2=1800$:

• Digit #$1$ can be any of the $ 9$ digits in $[ 1,2,3,4,5,6,7,8,9]$
• Digit #$2$ can be any of the $10$ digits in $[0,1,2,3,4,5,6,7,8,9]$
• Digit #$3$ can be any of the $10$ digits in $[0,1,2,3,4,5,6,7,8,9]$
• Digit #$4$ can be any of the $ 2$ digits in $[0, 5 ]$

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$3\cdot10\cdot10\cdot5=1500$:

• Digit #$1$ can be any of the $ 3$ digits in $[ 1,2,3 ]$
• Digit #$2$ can be any of the $10$ digits in $[0,1,2,3,4,5,6,7,8,9]$
• Digit #$3$ can be any of the $10$ digits in $[0,1,2,3,4,5,6,7,8,9]$
• Digit #$4$ can be any of the $ 5$ digits in $[ 1, 3, 5, 7, 9]$