Question

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions 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 1

No of 5 digit numbers starting with digit 1 $=5 \times 5 \times 5 \times 5=625$
No of 5 digit numbers starting with digit 2 $=5 \times 5 \times 5 \times 5=625$
No of 5 digit numbers starting with 31 $=5 \times 5 \times 5=125$
No of 5 digit numbers starting with 32 $=5 \times 5 \times 5=125$
No of 5 digit numbers starting with 33 $=5 \times 5 \times 5=125$
No of 5 digit numbers starting with 351 $=5 \times 5=25$
No of 5 digit numbers starting with 352 $=5 \times 5=25$
No of 5 digit numbers starting with 3531 $=5$
No of 5 digit numbers starting with 3532 $=5$
Before 35337 will be 4 numbers,
So rank of 35337 will be 1690
So, in descending order serial number will be $3125-1690+1=1436$