Question is:
If we have $1$, $2$, $3$, $4$, $5$. Find how many even number greater than
$300$ can be formed from these digits, if the digits:(A) Can be repeated
(B) Can not be repeatedAnswers given (A) $1530$, (B) $111$
My answer is like:
(A) $(5^4 \times 2)+(5^3\times2)+(3\times5\times2)=1530$
(B) $(1\times2\times3\times4\times2)+(2\times3\times4\times2)+\mathbf{(3\times3\times2)}= 114$
I believe the part in bold is where the problem is so I will explain how I went about it:
I have three slots _ _ _ the last must be $2$ or $4$ (even) so $2$ possible choices and the first can be $3, 4, 5$ so $3$ choices and the middle can be anything except the past $2$ choices so $5-2=3$ choices so $3\times3\times2$.
I answered the A part using slot method but whenever I try the B part I get $114$ as an answer.
What am I doing wrong and how can I fix it?
Best Answer
For three digit numbers, when you have a 2 at the end, this works. When you have a 4 at the end, it does not. Instead of $3\cdot 3\cdot 2$, you have $3\cdot 3\cdot 1 + 2\cdot 3\cdot 1$ which is precisely 3 fewer than what you found. You cannot repeat the 4.