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Let's work through the problem step by step:

### Problem Setup:
Sheetal's password consists of:
- 2 different vowels
- 2 different consonants
- 1 numerical digit

We are asked to use the concepts of permutations and combinations.

### Part (i): **Find the total number of passwords she needs to try.**

1. **Vowels**: There are 5 vowels (A, E, I, O, U). We need to choose 2 different vowels.
   - Number of ways to choose 2 vowels = \( \binom{5}{2} = \frac{5 \times 4}{2} = 10 \).

2. **Consonants**: There are 21 consonants in the English alphabet. We need to choose 2 different consonants.
   - Number of ways to choose 2 consonants = \( \binom{21}{2} = \frac{21 \times 20}{2} = 210 \).

3. **Digits**: She uses one digit (0–9), so there are 10 possible choices.
   - Number of ways to choose 1 digit = 10.

4. **Arrangements**: Now, we have 5 characters (2 vowels, 2 consonants, and 1 digit), and we need to arrange these 5 characters. The number of ways to arrange 5 items is \(5! = 120\).

Thus, the total number of possible passwords is:
\[
10 \times 210 \times 10 \times 120 = 252000.
\]

---

### Part (ii): **If she remembers that the numerical digit 0 is in her password and the password starts with S, how many passwords should she try?**

1. **Vowels and consonants**: Since S is fixed at the start and digit 0 is used, we are left with:
   - 2 vowels from the 5 available vowels.
   - 1 consonant from the remaining 20 consonants (since S is already chosen).
   
   The number of ways to choose 2 vowels = \( \binom{5}{2} = 10 \).  
   The number of ways to choose 1 consonant = \( \binom{20}{1} = 20 \).

2. **Arrangements**: Now we need to arrange the 3 characters (2 vowels and 1 consonant) and place digit 0 somewhere in the remaining 4 positions (since the first position is occupied by S). This gives \( 4! = 24 \) arrangements.

Thus, the total number of passwords is:
\[
10 \times 20 \times 24 = 4800.
\]

---

### Part (iii): **If her password starts with S and ends with digit 0, how many passwords should she try?**

1. **Vowels and consonants**: Similar to part (ii), but now both S and 0 are fixed, leaving us with:
   - 2 vowels from 5 vowels.
   - 1 consonant from 20 remaining consonants.

2. **Arrangements**: The middle 3 characters (2 vowels and 1 consonant) can be arranged in \( 3! = 6 \) ways.

Thus, the total number of passwords is:
\[
10 \times 20 \times 6 = 1200.
\]

---

### Part (iv): **If she makes new passwords by arranging all the letters of her name, how many passwords are possible with both Es coming together?**

1. **Sheetal's name**: The letters are S, H, E, E, T, A, L (7 letters in total).

2. **Both Es together**: Consider both Es as a single entity. Now, we are arranging 6 items (SH, T, A, L, and the "EE" block).

   The number of ways to arrange 6 items = \( 6! = 720 \).

Thus, the total number of passwords where both Es come together is 720.

---

Would you like any more detailed explanations or have any questions?

Here are 5 relative questions for you to explore:

1. How would the total change if both consonants must be the same?
2. What if the password must include a digit other than 0?
3. How does fixing two characters in specific positions (e.g., start and end) affect password possibilities?
4. Can you derive the formula for calculating arrangements when certain characters must be together?
5. What if she had used 3 different vowels instead of 2?

**Tip**: When counting passwords or combinations, breaking the problem into distinct parts and using combinations followed by arrangements simplifies complex scenarios.

Sheetal purchased a new laptop and set a password using 2 different vowels, 2 different consonants, and 1 numerical digit. But she forgot the password and does not want to reset it. She uses the concept of permutation and combination.

(i) Find the total number of passwords she needs to try?
(ii) If she remembers that she uses the numerical digit 0 in her password and password starts with S, then how many passwords should she try?
(iii) If her password starts with S and ends with digit 0 then how many passwords should she try?
(iv) If she makes new passwords by arranging all the letters of her name how many passwords are possible with both Es coming together?

Learn how to calculate password combinations involving vowels, consonants, and digits using permutations and combinations. Explore different cases such as starting with a specific letter or ending with a digit, and calculating passwords with specific letter arrangements. Suitable for Grades 10-12.

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