From the digits 1, 0, 2, 3, 5 and 6, how many different five digit numbers can be formed that are between 50000 and 60000, if no digit is repeated within a number?

Introduction / Context:
This problem is a digit arrangement question with a range restriction. We must form five digit numbers from a given set of digits, obeying two key conditions: the number falls within a specific interval and digits are not repeated. Such questions test whether students can correctly apply constraints on leading digits while still using permutations for the remaining positions.

Given Data / Assumptions:

• Available digits: 1, 0, 2, 3, 5 and 6.
• We must form five digit numbers.
• Digits cannot be repeated within a number.
• The number must be between 50000 and 60000, inclusive of 50000 and exclusive of 60000.

Concept / Approach:
A five digit number lies between 50000 and 60000 if and only if its first digit (ten thousands place) is 5. Thus, the leading digit is fixed. After fixing the first digit, we simply need to permute the remaining digits in the remaining positions, taking care not to repeat any digit. Since zero is allowed in positions other than the first, there is no restriction on placing 0 among the last four digits.

Step-by-Step Solution:
Step 1: Fix the first digit as 5, because only numbers starting with 5 are between 50000 and 60000 when using these digits.Step 2: After using 5, the remaining digits available for the last four places are {0, 1, 2, 3, 6}, which are five distinct digits.Step 3: We need to form all possible four digit sequences from these 5 digits without repetition for the remaining positions.Step 4: The number of ways to choose and arrange 4 positions from 5 distinct digits is 5P4 = 5 × 4 × 3 × 2.Step 5: Compute 5 × 4 × 3 × 2 = 120.Step 6: Each such arrangement corresponds to a unique five digit number starting with 5 and hence lying between 50000 and 60000.
Verification / Alternative check:
We can check that there is no additional condition affecting the last four digits. Since zero is not allowed as the first digit but is freely permitted in other positions, our count is exhaustive. Also, if we tried an alternative count by first choosing which digit is left unused and then arranging the others, we would get the same total: 5 ways to choose the omitted digit times 4! ways to arrange the remaining digits, giving 5 × 24 = 120.

Why Other Options Are Wrong:
Values like 240 or 360 would arise if we mistakenly allowed repetition of digits or misapplied permutation formulas. 60 would correspond to halving the correct count, which might happen if one incorrectly restricts the positions of zero. Only 120 correctly reflects the number of valid five digit numbers satisfying all conditions.

Common Pitfalls:
A common mistake is to forget that the first digit is forced to be 5 by the given range, and instead treat all digits as freely permutable. Another error is misplacing restrictions on zero and assuming it cannot appear at all, which would unnecessarily reduce the count. Careful consideration of each place value and the stated range prevents these errors.

Final Answer:
The number of five digit numbers between 50000 and 60000 that can be formed is 120.