2007 Cyprus MO/Lyceum/Problem 19

Problem

120 five-digit numbers can be written with the digits $1,2,3,4,5$ . If we place these numbers in increasing order, then the position of the number $41253$ is

$\mathrm{(A) \ } 71\qquad \mathrm{(B) \ } 72\qquad \mathrm{(C) \ } 73\qquad \mathrm{(D) \ } 74\qquad \mathrm{(E) \ } \mathrm{None\ of\ these}$

Solution

For any number in the first slot, there are $4! = 24$ numbers, so any number starting with $4$ must greater than $3 * 24 = 72$ . After than, the rearrangement inequality gives us that $1,\ 2$ are in order, but $\displaystyle 41235 < 41253$ , so $41253$ is the $73 + 1 = 74 \Longrightarrow \mathrm{D}$ .

2007 Cyprus MO, Lyceum (Problems)
Preceded by Problem 18	Followed by Problem 20
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2007 Cyprus MO/Lyceum/Problem 19

Problem

Solution

See also