Competition problem — Combinatorics

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Combinatorics Difficulty 2.7 Junior Find the answer

There are twenty-four $4$ -digit numbers that use each of the four digits $2$ , $4$ , $5$ , and $7$ exactly once. Listed in numerical order from smallest to largest, the number in the $17\text{th}$ position in the list is

Pick one

Solution

For each choice of the thousands digit, there are $6$ numbers with that as the thousands digit. Thus, the six smallest are in the two thousands, the next six are in the four thousands, and then we need $5$ more numbers.
We can just list from here: $5247,5274,5427,5472,5724 \rightarrow \boxed{\text{B}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.