Question

The number of integers greater than 2000 that can be formed with the digits 0,1,2,3,4,5 using each digit at most once,is

• A. 1440
• B. 1200
• C. 1420
• D. 1480

Answer 1

The Correct Option is A

The correct answer is A:1440
We are given the digits 0,1,2,3,4, and 5,and we need to form integers greater than 2000 using these digits, with each digit being used at most once. 
Case 1: Integers between 2000 and 2999 
For the thousands place, we can choose any of the 6 digits except 0(0 cannot be the first digit).So, there are 5 choices. 
For the hundreds place, we have 5 remaining digits to choose from (since one digit has already been used in the thousands place). 
For the tens place, we have 4 remaining digits to choose from. 
For the units place, we have 3 remaining digits to choose from. 
Total integers in this range=\((5 \times 5 \times 4 \times 3 = 300) integers\). 
Case 2: Integers between 3000 and 4999 
For the thousands place, we can choose any of the 4 digits other than 0(since 0 cannot be the first digit anymore). 
For the hundreds place, we have 5 remaining digits to choose from. 
For the tens place, we have 4 remaining digits to choose from. 
For the units place, we have 3 remaining digits to choose from. 
Total integers in this range=\((4 \times 5 \times 4 \times 3 = 240) integers\). 
Case 3: Integers between 5000 and 5999 
For the thousands place, we can choose any of the 2 digits other than 0. 
For the hundreds place, we have 5 remaining digits to choose from. 
For the tens place, we have 4 remaining digits to choose from. 
For the units place, we have 3 remaining digits to choose from. 
Total integers in this range=\((2 \times 5 \times 4 \times 3 = 120) integers\). 
Total: Adding up the integers from all three ranges:(300+240+120=660) integers. 

However, we need to consider that there are 6 different digits available, and we can arrange them in \((6!)\) ways.
 This includes repetitions, which we need to exclude. So, the final answer is \((6!-660=1440)\) integers. 
Hence, the correct answer is 1440.

Answer 2

Three scenarios are possible. 

Case 1: Four-digit values without repetition that begin with 2, 3, 4, or 5 equal \(4 \times 5\times 4 \times3 = 240.\)


Case 2: Five consecutive digit numbers equal \(5 \times 5\times 4 \times 3 \times 2 = 600 \)

Case 3: Six consecutive digit digits equal \(5 \times 5\times 4 \times 3 \times 2 \times 1 = 600 \)

Total: 1440 (600 + 600 + 240).