Using the digits 2, 3, 4, 5 and 6 without repetition, how many distinct numbers greater than 4000 can be formed, considering both 4 digit and 5 digit numbers?

Introduction / Context:
This question examines counting of numbers formed from given digits without repetition, under a constraint that the overall number must be greater than 4000. It combines ideas of permutations with positional constraints on the leading digit and requires careful separation of cases for 4 digit and 5 digit numbers.

Given Data / Assumptions:

• Available digits are 2, 3, 4, 5 and 6.
• No digit may be repeated in any number.
• We are interested in all numbers greater than 4000.
• Both 4 digit and 5 digit numbers are allowed.
• No leading zero issue arises because 0 is not among the digits.

Concept / Approach:
We count separately the number of valid 4 digit numbers greater than 4000 and the number of valid 5 digit numbers. For 4 digit numbers, we enforce a restriction on the thousands place and then arrange remaining digits. For 5 digit numbers formed with all five digits without repetition, every such number is automatically greater than 4000 because the smallest possible leading digit is 2 which still produces a 5 digit number far above 4000.

Step-by-Step Solution:
Step 1: Count 4 digit numbers greater than 4000. The thousands digit must be 4, 5 or 6.Step 2: If the leading digit is fixed as 4, the remaining three places can be filled by any permutation of the remaining 4 digits (2,3,5,6), which gives 4P3 = 4 * 3 * 2 = 24 numbers.Step 3: Similarly, if the leading digit is 5, the remaining three places again have 4P3 = 24 possibilities.Step 4: If the leading digit is 6, the remaining three places again have 24 possibilities.Step 5: Total 4 digit numbers greater than 4000 = 24 + 24 + 24 = 72.Step 6: Now count 5 digit numbers. Using all five digits without repetition gives 5! = 120 distinct 5 digit numbers, and every one of them is greater than 4000.Step 7: Total valid numbers = 72 (4 digit) + 120 (5 digit) = 192.

Verification / Alternative check:
An alternative quick check is to list a smaller example with fewer digits and see the pattern. For instance, if digits were 3, 4, 5 only, you could test how many numbers above a threshold are formed and observe that splitting into cases based on the leading digit gives the correct count. Applying the same logic with five digits matches the formula based approach above. Also, reviewing the counts 72 and 120 separately and confirming that no case is double counted or missed builds confidence in the result.

Why Other Options Are Wrong:

• 120: This counts only the 5 digit permutations and ignores the 4 digit numbers greater than 4000.
• 256 and 244: These numbers are larger than the correct count and likely come from incorrect use of powers such as 4^4 or from including numbers with repetition, which is not allowed.

Common Pitfalls:
Some learners forget that 5 digit numbers automatically satisfy the greater than 4000 condition here and therefore undercount. Others accidentally allow repetition of digits or forget to restrict the thousands digit for 4 digit numbers. Another common mistake is to treat 4 digit and 5 digit cases together without careful separation, which can lead to double counting or miscounting. Always split into clear cases when constraints differ between number lengths.

Final Answer:
The total number of distinct numbers greater than 4000 that can be formed is 192.