Problem 6:
The three numbers $1,2,3$ are used to make a $5$ digit number. The five digit number must contain at least one $1$, at least one $2$, and at least one $3$. How many such five digit numbers can be made? (Hint: First count the number of words missing either a $1$ or a $2$ or a $3$.)
BdMO National Higher Secondary 2008/6
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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nafistiham
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Re: BdMO National Higher Secondary 2008/6
first, let us find how many combinations we can make with $1,2,3$
it is $3^5$
then we discount the combinations made with just $1,2;2,3;3,1$
the number is $3 \cdot 2^5$
but we have discounted the numbers $11111,22222,33333$ twice.so we have to add them again.
so the final number combination is
\[3^5-3 \cdot 2^5 +3\]
it is $3^5$
then we discount the combinations made with just $1,2;2,3;3,1$
the number is $3 \cdot 2^5$
but we have discounted the numbers $11111,22222,33333$ twice.so we have to add them again.
so the final number combination is
\[3^5-3 \cdot 2^5 +3\]
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.