Evaluate the expression.
StartFraction 17 Upper P 11 Over 11 exclamation mark EndFraction minus 17 Upper C 11


We need to evaluate the following expression:

\[
\frac{17P11}{11!} - 17C11
\]

where \( 17P11 \) and \( 17C11 \) represent the permutations and combinations respectively.

### Step 1: Definition of Permutation and Combination
- **Permutation** (\( 17P11 \)): The number of ways to arrange 11 objects out of 17. The formula for permutations is:

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

In this case, \( n = 17 \) and \( k = 11 \), so:

\[
17P11 = \frac{17!}{(17-11)!} = \frac{17!}{6!}
\]

- **Combination** (\( 17C11 \)): The number of ways to choose 11 objects out of 17 without regard to the order. The formula for combinations is:

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

So:

\[
17C11 = \frac{17!}{11!(17-11)!} = \frac{17!}{11!6!}
\]

### Step 2: Simplifying the Expression

We can now substitute the formulas:

\[
\frac{17P11}{11!} = \frac{\frac{17!}{6!}}{11!} = \frac{17!}{6! \cdot 11!}
\]

Thus, the expression becomes:

\[
\frac{17!}{6! \cdot 11!} - \frac{17!}{11! \cdot 6!}
\]

Notice that both terms are exactly the same. So:

\[
\frac{17P11}{11!} - 17C11 = 0
\]

### Final Answer:
The value of the expression is \( \boxed{0} \).

Would you like a more detailed explanation, or do you have any further questions? Here are some related questions:

1. What is the relationship between permutations and combinations?
2. How do we simplify factorials in mathematical expressions?
3. What is the difference between arranging objects and choosing objects?
4. How do you calculate large factorials for combinations or permutations?
5. How do factorials relate to probability and statistics?

Tip: In many cases, when you see terms like \( n! \) in both the numerator and denominator, you can cancel out common factorials to simplify the expression before evaluating.

Evaluate the expression. StartFraction 17 Upper P 11 Over 11 exclamation mark EndFraction minus 17 Upper C 11

Learn how to evaluate the expression involving permutations and combinations, 17P11 / 11! - 17C11. The solution explains the steps to simplify the expression, using permutation and combination formulas and factorial simplifications. Suitable for students in Grades 10-12.

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