How many Numbers are possible in the cases given below from the digits 1, 2, 3, 4, 5 ? Solve with an (without Repetition only)
(c) Greater than 35

To solve the problem of finding how many numbers can be formed from the digits \(1, 2, 3, 4, 5\) without repetition and where the number is greater than 35, we will break it down step by step.

### Step-by-Step Approach

We are dealing with digits \(1, 2, 3, 4, 5\) and want to find the number of possible **2-digit** numbers greater than 35. This can be solved by considering cases where the tens digit must be either 4 or 5, as these are the only cases where the number will exceed 35.

1. **Tens place:**
   - The tens digit must be either **4** or **5** for the number to be greater than 35.

2. **Units place:**
   - Once a digit is chosen for the tens place, we have 4 remaining digits available for the units place (since repetition is not allowed).

#### Case 1: Tens digit is 4
- If the tens digit is 4, the remaining available digits for the units place are \(1, 2, 3, 5\). Therefore, there are 4 choices for the units place.

#### Case 2: Tens digit is 5
- If the tens digit is 5, the remaining available digits for the units place are \(1, 2, 3, 4\). Therefore, there are also 4 choices for the units place.

### Total Number of Possible Numbers

For both cases, the number of possible numbers can be calculated by adding the possible outcomes:

- **Case 1 (Tens = 4)**: 4 choices for the units place.
- **Case 2 (Tens = 5)**: 4 choices for the units place.

Thus, the total number of 2-digit numbers greater than 35 is:

\[
4 + 4 = 8
\]

### Conclusion
There are **8** possible numbers that can be formed from the digits \(1, 2, 3, 4, 5\) that are greater than 35 without repetition.

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Would you like further details on this or have any additional questions? Here are five related questions for further exploration:

1. How many 2-digit numbers less than 35 can be formed with the digits \(1, 2, 3, 4, 5\) without repetition?
2. How many total 2-digit numbers can be formed with the digits \(1, 2, 3, 4, 5\) without repetition, regardless of the value?
3. What if repetition of digits was allowed, how would the number of possible outcomes change?
4. How many 3-digit numbers greater than 135 can be formed using the digits \(1, 2, 3, 4, 5\) without repetition?
5. How would the number of possibilities change if we included zero as an available digit?

**Tip:** When solving counting problems with restrictions, it’s often helpful to break the problem into cases that meet the restriction.

How many Numbers are possible in the cases given below from the digits 1, 2, 3, 4, 5 ? Solve with an (without Repetition only) (c) Greater than 35

Solve how many 2-digit numbers greater than 35 can be formed using digits 1, 2, 3, 4, 5 without repetition. Explore concepts of combinatorics, inequalities, and the fundamental principle of counting. Suitable for students in Grades 6-8.

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