Evaluate the following expression. Express your answer as a fraction or a decimal number rounded to four decimal places.

11C7/11P3

To evaluate the expression \( \frac{{11C7}}{{11P3}} \), we need to first compute \( 11C7 \) and \( 11P3 \).

### Step 1: Compute \( 11C7 \)
The combination formula \( nCk \) is given by:

\[
nCk = \frac{{n!}}{{k!(n-k)!}}
\]

For \( 11C7 \), we substitute \( n = 11 \) and \( k = 7 \):

\[
11C7 = \frac{{11!}}{{7!(11-7)!}} = \frac{{11!}}{{7!4!}}
\]

Now, calculating the factorials:

\[
11! = 11 \times 10 \times 9 \times 8 \times 7!
\]

The \( 7! \) cancels out:

\[
11C7 = \frac{{11 \times 10 \times 9 \times 8}}{{4!}} = \frac{{11 \times 10 \times 9 \times 8}}{{24}} = \frac{{7920}}{{24}} = 330
\]

So, \( 11C7 = 330 \).

### Step 2: Compute \( 11P3 \)
The permutation formula \( nPk \) is given by:

\[
nPk = \frac{{n!}}{{(n-k)!}}
\]

For \( 11P3 \), we substitute \( n = 11 \) and \( k = 3 \):

\[
11P3 = \frac{{11!}}{{(11-3)!}} = \frac{{11!}}{{8!}}
\]

Now, calculating the factorials:

\[
11! = 11 \times 10 \times 9 \times 8!
\]

The \( 8! \) cancels out:

\[
11P3 = 11 \times 10 \times 9 = 990
\]

So, \( 11P3 = 990 \).

### Step 3: Compute the final expression
Now we substitute the values of \( 11C7 \) and \( 11P3 \):

\[
\frac{{11C7}}{{11P3}} = \frac{{330}}{{990}} = \frac{1}{3} \approx 0.3333
\]

Thus, the value of the expression is \( \frac{1}{3} \) or \( 0.3333 \).

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Do you want more details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the difference between combinations and permutations?
2. How would the result change if the expression was \( \frac{12C8}{12P4} \)?
3. Can you explain how factorials play a role in counting combinations and permutations?
4. What is \( 11C4 \) and how does it relate to \( 11C7 \)?
5. How can you generalize the combination and permutation formulas for any \( n \) and \( k \)?

**Tip:** When working with combinations and permutations, simplify factorials by canceling common terms to make calculations easier!

Evaluate the following expression. Express your answer as a fraction or a decimal number rounded to four decimal places. 11C7/11P3

Learn how to evaluate the expression 11C7/11P3 with a detailed step-by-step solution. This problem involves combinations, permutations, and factorials, providing key insights into combinatorial mathematics. Suitable for students in Grades 10-12.

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