Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetition, and are writ…

Question

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetition, and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is \_\_\_\_\_\_\_.

Answer: 1436

Accepted Answer

Numbers starting with 7: $5^4=625$; with 5: $625$ (cumulative $1250$); starting with 3 comes next.

Within "3xxxx": starting with 31: $125$; 32: $125$; then "33xxx" ($125$ total): starting with 331,332: $25$ each; then within "333xx" --- continuing digit by digit for "35337":

Numbers with first digit $>3$ (i.e., $7,5$): $625+625=1250$. Numbers starting with "3" and second digit $>5$ (i.e. 7): $125$ (cumulative $1375$). Numbers starting "35" with third digit $>3$ (i.e. 5,7): $2\times25=50$ (cumulative $1425$). Numbers starting "353" with fourth digit $>3$: $2\times5=10$ (cumulative $1435$). Then "3533" with fifth digit $\ge7$: $1$ (cumulative $1436$), landing exactly at $35337$.

$$\text{Serial number of } 35337 = 1436$$

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