Convert the following into factorial.
(i) 2.4.6.8.10.12.14.16.18.20
(ii) 1.3.5.7.9.11.13.15
(iii) 3.6.9.12.15
(iv) 6.7.8.9.10

Answer

To convert the given products into factorials, we will identify the missing numbers in each sequence and express the product as a fraction of factorials.

### Solution Steps:

**(i) Convert 2 × 4 × 6 × 8 × 10 × 12 × 14 × 16 × 18 × 20 into factorial:**

1. **Identify the sequence:** The numbers are all even numbers from 2 to 20.
2. **Find the missing numbers:** The odd numbers from 1 to 19 are missing: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
3. **Combine the sequences:** The product can be expressed as:
\[ 2 × 4 × 6 × 8 × 10 × 12 × 14 × 16 × 18 × 20 = \frac{20!}{1 × 3 × 5 × 7 × 9 × 11 × 13 × 15 × 17 × 19} \]
4. **Calculate the denominator:** The product of the odd numbers (1, 3, 5, 7, 9, 11, 13, 15, 17, 19) is 10395.

Thus, we have:
\[ 2 × 4 × 6 × 8 × 10 × 12 × 14 × 16 × 18 × 20 = \frac{20!}{10395} \]

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**(ii) Convert 1 × 3 × 5 × 7 × 9 × 11 × 13 × 15 into factorial:**

1. **Identify the sequence:** The numbers are all odd numbers from 1 to 15.
2. **Find the missing numbers:** The even numbers from 2 to 14 are missing: 2, 4, 6, 8, 10, 12, 14.
3. **Combine the sequences:** The product can be expressed as:
\[ 1 × 3 × 5 × 7 × 9 × 11 × 13 × 15 = \frac{15!}{2 × 4 × 6 × 8 × 10 × 12 × 14} \]
4. **Calculate the denominator:** The product of the even numbers (2, 4, 6, 8, 10, 12, 14) is 645120.

Thus, we have:
\[ 1 × 3 × 5 × 7 × 9 × 11 × 13 × 15 = \frac{15!}{645120} \]

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**(iii) Convert 3 × 6 × 9 × 12 × 15 into factorial:**

1. **Identify the sequence:** The numbers are multiples of 3 up to 15.
2. **Find the missing numbers:** The numbers 1, 2, 4, 5, 7, 8, 10, 11, 13, 14 are missing.
3. **Combine the sequences:** The product can be expressed as:
\[ 3 × 6 × 9 × 12 × 15 = \frac{15!}{1 × 2 × 4 × 5 × 7 × 8 × 10 × 11 × 13 × 14} \]
4. **Calculate the denominator:** The product of the missing numbers is 2222400.

Thus, we have:
\[ 3 × 6 × 9 × 12 × 15 = \frac{15!}{2222400} \]

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**(iv) Convert 6 × 7 × 8 × 9 × 10 into factorial:**

1. **Identify the sequence:** The numbers are consecutive integers from 6 to 10.
2. **Find the missing numbers:** The numbers 1, 2, 3, 4, 5 are missing.
3. **Combine the sequences:** The product can be expressed as:
\[ 6 × 7 × 8 × 9 × 10 = \frac{10!}{1 × 2 × 3 × 4 × 5} \]
4. **Calculate the denominator:** The product of the missing numbers is 120.

Thus, we have:
\[ 6 × 7 × 8 × 9 × 10 = \frac{10!}{120} \]

### Final Results:

1. \( 2 × 4 × 6 × 8 × 10 × 12 × 14 × 16 × 18 × 20 = \frac{20!}{10395} \)
2. \( 1 × 3 × 5 × 7 × 9 × 11 × 13 × 15 = \frac{15!}{645120} \)
3. \( 3 × 6 × 9 × 12 × 15 = \frac{15!}{2222400} \)
4. \( 6 × 7 × 8 × 9 × 10 = \frac{10!}{120} \)