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Let's solve the two parts using counting principles.

### Part (a): 
The letters of the word "TUESDAY" are arranged, each arrangement ending with the letter S and starting with the letter D. The word "TUESDAY" has 7 distinct letters: T, U, E, S, D, A, Y.

To solve this, we need to fix two positions:
- The first letter must be **D**.
- The last letter must be **S**.

Now we have 5 remaining letters: T, U, E, A, Y, and we need to arrange these 5 letters in the middle.

So, the number of possible ways to arrange the remaining letters is simply the number of permutations of these 5 letters, which is:
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

Thus, the total number of ways to arrange the letters of the word "TUESDAY" with the given conditions is **120**.

### Part (b):
We need to count how many numbers greater than 40,000 can be formed using the digits 2, 3, 4, 5, and 6, with each digit used only once.

#### Step 1: Determine the total number of digits available.
The digits are 2, 3, 4, 5, and 6. We are looking for 5-digit numbers where each digit can only be used once.

#### Step 2: Condition for the number being greater than 40,000.
For a number to be greater than 40,000, the first digit must be either 4, 5, or 6 (since choosing 2 or 3 as the first digit would result in numbers less than 40,000).

#### Step 3: Calculate the number of possibilities.
- **Case 1**: First digit is 4. The remaining digits are 2, 3, 5, and 6. These can be arranged in \(4!\) ways:
  \[
  4! = 24
  \]
- **Case 2**: First digit is 5. The remaining digits are 2, 3, 4, and 6. These can also be arranged in \(4!\) ways:
  \[
  4! = 24
  \]
- **Case 3**: First digit is 6. The remaining digits are 2, 3, 4, and 5. These can be arranged in \(4!\) ways:
  \[
  4! = 24
  \]

Thus, the total number of numbers greater than 40,000 is:
\[
24 + 24 + 24 = 72
\]

### Final Answers:
- Part (a): 120 ways.
- Part (b): 72 numbers.

Would you like further details or clarifications on any part? 

### Related questions:
1. How would the answer change in part (a) if some of the letters were repeated?
2. How many numbers greater than 30,000 can be formed using the digits 1, 2, 3, 4, 5, and 6?
3. In part (a), what if the arrangement should start with D and end with Y?
4. How many numbers can be formed using the digits 2, 3, 4, 5, and 6 without any restrictions?
5. In part (b), what if the number must be even?

### Tip:
For counting problems, always identify any fixed or restricted positions first, as this simplifies the calculation considerably.

a) The letters of the word TUESDAY are arranged in a line, each arrangement ending with the letter S and starting with the letter D. In how many ways is this possible?
b) How many numbers greater than 40000 can be formed using the digits 2, 3, 4, 5, and 6 if each digit is used only once in each number?

Learn how to solve combinatorics problems involving arranging the letters of the word 'TUESDAY' with specific restrictions, and how to form numbers greater than 40,000 using the digits 2, 3, 4, 5, and 6. This problem involves key counting principles and permutation calculations, suitable for students in Grades 9-12.

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