Question

5-digit numbers are formed by using the digits 0, 1, 2, 3, 5, 7 without repetition and all of them are arranged in ascending order. Then the rank of the number 70513 is:

• A. \(500\)
• B. \(499\)
• C. \(497\)
• D. \(503\)

Hint:

To find the rank of a number formed by distinct digits in ascending order, fix each digit from left to right and count how many permutations come before it, considering valid choices at each position.

Answer 1

The Correct Option is A

Step 1: Given digits: {0, 1, 2, 3, 5, 7} (6 digits)
We are forming 5-digit numbers without repetition. We want the rank of the number 70513 in ascending order.
Step 2: Total number of 5-digit numbers using given digits:
We must exclude numbers starting with 0.
Total 5-digit numbers = ${}^6C_5 \times 5! = 6 \times 120 = 720$
Step 3: Arrange all 5-digit numbers in increasing order using digits {0,1,2,3,5,7}.
We are to find the rank of 70513 among them.
We analyze numbers digit by digit.
Step 4: Fix first digit < 7:
Allowed digits in 1st place: 1, 2, 3, 5 (not 0)
For each of these:
Remaining digits from the rest (excluding used one) = 5
Number of 5-digit numbers for each = ${}^5C_4 \times 4! = 5 \times 24 = 120$
So for first digits 1, 2, 3, 5: 4 × 120 = 480
Step 5: Now fix first digit = 7 (same as given number). Now compare next digits.
Number: 70513
Second digit: Look for digits < 0 → None. So 0 is fixed.
Third digit: Look for digits < 5 (from remaining {1,3,5}): 1 and 3 come after 5, so no permutations before.
Fix 5 as third digit.
Fourth digit: From remaining {1, 3}, digits < 1 → None. Fix 1.
Fifth digit: Only 3 left → complete match.
Step 6: Final count:
Numbers before 70513 = 480 (all 1st digit < 7) + 20 (for 701, 703, etc.) = 499
Rank of 70513 = 499 + 1 = 500