JEE MainMathematics

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Permutations and Combinations: Mathematics | JEE Main

Q: Why does map to and not ?

Because is the first number (serial number 1), so it must map to 0 in base-5 so that .

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

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Step-by-step solutionView

Correct answer

The serial number of 35337 is 1436.

Option analysis

Why each option works or fails

Reviewed route

Solution

StepWorking

01given

Digits available: $\{1, 2, 3, 5, 7\}$ (5 digits). Repetition is allowed. Numbers are arranged in descending order starting from $77777$ at serial number 1. Target number is $35337$ .

02goal

Find the serial number (rank) of the number $35337$ in this descending order list.

03approach

Count all 5-digit numbers strictly greater than $35337$ using the available digits in descending priority (7, 5, 3, 2, 1). The serial number of $35337$ will be $(\text{count of numbers strictly greater}) + 1$ .

04execute

Count numbers starting with digits strictly greater than the first digit 3: - Numbers starting with 7: $1 \times 5^4 = 625$ - Numbers starting with 5: $1 \times 5^4 = 625$

05execute

First digit is 3. Now count numbers with second digit strictly greater than 5: - Numbers starting with $37$ : $1 \times 1 \times 5^3 = 125$

06execute

First two digits are 35. Count numbers with third digit strictly greater than 3: - Numbers starting with $357$ : $1 \times 5^2 = 25$ - Numbers starting with $355$ : $1 \times 5^2 = 25$

07execute

First three digits are 353. Count numbers with fourth digit strictly greater than 3: - Numbers starting with $3537$ : $1 \times 5^1 = 5$ - Numbers starting with $3535$ : $1 \times 5^1 = 5$

08execute

First four digits are 3533. Fifth digit in descending order: 7 is the first and largest possible digit. So no numbers strictly greater than 35337 begin with 3533. Sum all strictly greater numbers and add 1 for 35337 itself: $\text{Rank} = 625 + 625 + 125 + 25 + 25 + 5 + 5 + 1 = 1436$

✓verify

Total numbers possible is $5^5 = 3125$ . The mapped base-5 digit string for $35337$ under descending bijection $(7\to 0, 5\to 1, 3\to 2, 2\to 3, 1\to 4)$ is $21220_5$ . Converting $21220_5$ to base 10 gives $2 \cdot 625 + 1 \cdot 125 + 2 \cdot 25 + 2 \cdot 5 + 0 = 1250 + 125 + 50 + 10 = 1435$ . Adding 1 for 1-based indexing gives $1436$ . The result is verified.

✓ Source and academic review

Question type: Numerical
Exam relevance: JEE Main · Mathematics
Concepts assessed: Mathematics
Academic status: Reviewed by official_key
Source: pyq
Editorial review: 8 September 2026

Quick checks

Students also ask

Why do we count numbers greater than 35337 instead of less than it?

Because the problem specifies that numbers are arranged in descending order, meaning 77777 is #1, followed by smaller numbers. Hence, the rank equals the count of strictly larger numbers plus 1.

Why does

7

map to

0

and not

4

Because $77777$ is the first number (serial number 1), so it must map to 0 in base-5 so that $0 + 1 = 1$ .

Answer

The serial number of 35337 is 1436.

Why each option works or fails

Step-by-step solution

given: Digits available: $\{1, 2, 3, 5, 7\}$ (5 digits). Repetition is allowed. Numbers are arranged in descending order starting from $77777$ at serial number 1. Target number is $35337$ .
goal: Find the serial number (rank) of the number $35337$ in this descending order list.
approach: Count all 5-digit numbers strictly greater than $35337$ using the available digits in descending priority (7, 5, 3, 2, 1). The serial number of $35337$ will be $(\text{count of numbers strictly greater}) + 1$ .
execute: Count numbers starting with digits strictly greater than the first digit 3: - Numbers starting with 7: $1 \times 5^4 = 625$ - Numbers starting with 5: $1 \times 5^4 = 625$
execute: First digit is 3. Now count numbers with second digit strictly greater than 5: - Numbers starting with $37$ : $1 \times 1 \times 5^3 = 125$
execute: First two digits are 35. Count numbers with third digit strictly greater than 3: - Numbers starting with $357$ : $1 \times 5^2 = 25$ - Numbers starting with $355$ : $1 \times 5^2 = 25$
execute: First three digits are 353. Count numbers with fourth digit strictly greater than 3: - Numbers starting with $3537$ : $1 \times 5^1 = 5$ - Numbers starting with $3535$ : $1 \times 5^1 = 5$
execute: First four digits are 3533. Fifth digit in descending order: 7 is the first and largest possible digit. So no numbers strictly greater than 35337 begin with 3533. Sum all strictly greater numbers and add 1 for 35337 itself: $\text{Rank} = 625 + 625 + 125 + 25 + 25 + 5 + 5 + 1 = 1436$
verify: Total numbers possible is $5^5 = 3125$ . The mapped base-5 digit string for $35337$ under descending bijection $(7\to 0, 5\to 1, 3\to 2, 2\to 3, 1\to 4)$ is $21220_5$ . Converting $21220_5$ to base 10 gives $1 \times 5^1 = 5$ . Adding 1 for 1-based indexing gives $1436$ . The result is verified.

Shortcut:

When to use it: When forming all combinations with repetition from an n-element set, isomorphic to base-n representation.

given: Alphabet $A = \{7, 5, 3, 2, 1\}$ arranged in descending order. Total 5 digits.

approach: Map descending symbols to base-5 digits: $7 \mapsto 0$ , $5 \mapsto 1$ , $3 \mapsto 2$ , $2 \mapsto 3$ , $1 \mapsto 4$ . Then serial number 1 corresponds to $00000_5 = 0$ . The serial number of any sequence is $(N)_{10} + 1$ .

execute: Map $35337$ : $3 \to 2$ , $5 \to 1$ , $3 \to 2$ , $3 \to 2$ , $7 \to 0$ . So the number corresponds to $21220_5$ . Value in base 10: $2 \times 5^4 + 1 \times 5^3 + 2 \times 5^2 + 2 \times 5^1 + 0 \times 5^0 + 1 = 1250 + 125 + 50 + 10 + 0 + 1 = 1436$ .

verify: $21220_5 + 1 = 1436$ . Checked arithmetic.

Permutations and Combinations: Mathematics | JEE Main

What feels right?

The serial number of 35337 is 1436.

Why each option works or fails

Solution

01given

02goal

03approach

04execute

05execute

06execute

07execute

08execute

✓verify

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