Questions: Evaluate 9C6 and 11P5.

Transcript text: Evaluate ${ }_{9} \mathrm{C}_{6}$ and ${ }_{11} \mathrm{P}_{5}$.

Solution

Solution Steps

To evaluate the combinations and permutations, we can use the formulas for combinations and permutations. For combinations, the formula is $ \binom{n}{k} = \frac{n!}{k!(n-k)!} $. For permutations, the formula is $ P(n, k) = \frac{n!}{(n-k)!} $. We will use Python's math module to compute the factorials and then apply these formulas.

Step 1: Evaluate Combinations

To evaluate $ _{9}C_{6} $, we use the formula for combinations:

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

Substituting $ n = 9 $ and $ k = 6 $:

\[ _{9}C_{6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6! \cdot 3!} \]

Calculating this gives:

\[ _{9}C_{6} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

Step 2: Evaluate Permutations

To evaluate $ _{11}P_{5} $, we use the formula for permutations:

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

Substituting $ n = 11 $ and $ k = 5 $:

\[ _{11}P_{5} = \frac{11!}{(11-5)!} = \frac{11!}{6!} \]