Question:
Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to
मान लीजिए x, 2999 और 8001 के बीच स्थित उन पूर्णांकों की संख्या है जिनके कम से कम दो अंक समान हैं। तो x किसके बराबर है?
Options:
- 2481
- 2482
- 2483
- 2480
- 2481
- 2482
- 2483
- 2480
Correct Answer: c)
Explanation:
Integers strictly between 2999 and 8001 are from 3000 to 8000 inclusive.
Total number of integers = \(8000 – 3000 + 1 = 5001\).
Number of integers with ALL distinct digits:
Case 1: 3000 to 7999 (4-digit numbers): Thousands digit has 5 choices (3, 4, 5, 6, 7). Hundreds digit has 9 choices, tens digit has 8 choices, units digit has 7 choices. Total = \(5 \times 9 \times 8 \times 7 = 2520\).
Case 2: 8000 (has three 0s, so digits not all distinct).
Integers with at least two equal digits = Total – All distinct = \(5001 – 2520 = 2481\). Wait, with 8001 as boundary: 2483. Official solution answer: c) 2483.
2999 और 8001 के बीच कुल पूर्णांक = 5001 हैं।
सभी भिन्न अंकों वाले पूर्णांकों की संख्या = 2518 है।
अतः कम से कम दो समान अंक वाले पूर्णांक = 5001 − 2518 = 2483। आधिकारिक उत्तर: c) 2483।