https://file.aiquickdraw.com/mathbot/file/17302296494784t5h9jp0.jpg

To solve the given expression, let’s break down each component.

The expression is:

\[
\frac{{^{12}P_7}}{{7!}} - {^{12}C_7}
\]

### Step 1: Calculate \( ^{12}P_7 \) (Permutations)
The formula for permutations is:
\[
^{n}P_r = \frac{n!}{(n - r)!}
\]
For \( ^{12}P_7 \):
\[
^{12}P_7 = \frac{12!}{(12 - 7)!} = \frac{12!}{5!}
\]

### Step 2: Calculate \( 7! \)
This will help simplify the fraction:
\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\]

### Step 3: Compute \( \frac{^{12}P_7}{7!} \)
We will evaluate \( 12! \), divide by \( 5! \), and then divide the result by \( 7! \).

### Step 4: Calculate \( ^{12}C_7 \) (Combinations)
The formula for combinations is:
\[
^{n}C_r = \frac{n!}{r!(n - r)!}
\]
For \( ^{12}C_7 \):
\[
^{12}C_7 = \frac{12!}{7! \times 5!}
\]

### Step 5: Substitute and Simplify
Once we have the values, we can substitute and solve the expression:

\[
\frac{^{12}P_7}{7!} - {^{12}C_7}
\]

I’ll calculate these values next to find the solution.The result of the expression is:

\[
\frac{{^{12}P_7}}{{7!}} - {^{12}C_7} = 0
\]

Would you like a more detailed explanation on any part of this calculation?

### Related Questions
1. How do permutations and combinations differ in their calculations?
2. Why do we divide by \( 7! \) in the expression?
3. How would the answer change if we used a different value for \( n \) or \( r \)?
4. What are other applications of permutations and combinations in probability?
5. How would you calculate \( ^{12}P_7 \) and \( ^{12}C_7 \) manually?

**Tip:** Remember that combinations ignore order, while permutations account for it.

Learn how to solve (12P7 / 7!) - 12C7 using permutation and combination formulas. This solution covers factorial calculations and is suitable for high school students studying advanced mathematics.

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