JEE Main 2023 · 24 January · evening · Single correct · +4 / −1
The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is :
- (a) 48
- (b) 120
- (c) 168
- (d) 220
Answer and solution
Answer (c) 168
Solution
Maths sir · 4 steps
Split by length: 4-digit numbers need a big first digit, and every 5-digit number already clears 7000.
- Which lengths count. With only five digits and no repetition, a number has at most 5 digits. A number with 3 or fewer digits is below 1000, so only 4-digit and 5-digit numbers can be greater than 7000.
- Four-digit numbers. To exceed 7000, the thousands digit must be 7 or 8, so there are 2 choices. The remaining three places are filled from the 4 unused digits in 4 × 3 × 2 = 24 ways, giving 2 × 24 = 48 numbers. None of these equals 7000, since 0 is not available.
- Five-digit numbers. Every 5-digit number is at least 10000, so all arrangements of the five digits work: 5! = 120 numbers.
48 + 120 = 168
- So. There are 168 such integers, which is option (c).
The idea to keep. For a "greater than N" count, first decide which lengths are possible. Restrict the leading digit only for numbers with the same length as N; every longer number counts in full.
Answer from the NTA final answer key. Solution by PrepWiser.