Evaluate $$ { }_{11} P_{5} $$ and $$ { }_{11} \mathrm{C}_{4} $$ \begin{tabular}{|}$$ { }_{11} P_{5}= $$ \ $$ { }_{11} C_{4}= $$ \ \end{tabular}

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UpStudy AI · Accepted Answer

To evaluate $$ {}_{11}P_{5} $$ (the number of permutations of 11 objects taken 5 at a time) and $$ {}_{11}C_{4} $$ (the number of combinations of 11 objects taken 4 at a time), we'll use the following formulas:

1. **Permutations:**
   $$
   {}_{n}P_{r} = \frac{n!}{(n - r)!}
   $$
   
2. **Combinations:**
   $$
   {}_{n}C_{r} = \frac{n!}{r!(n - r)!}
   $$

Given $$ n = 11 $$:

### 1. Calculating $$ {}_{11}P_{5} $$:
$$
{}_{11}P_{5} = \frac{11!}{(11 - 5)!} = \frac{11!}{6!}
$$
Breaking it down:
$$
11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6!
$$
So,
$$
{}_{11}P_{5} = 11 \times 10 \times 9 \times 8 \times 7 = 55,\!440
$$

### 2. Calculating $$ {}_{11}C_{4} $$:
$$
{}_{11}C_{4} = \frac{11!}{4! \times 7!}
$$
Simplifying the factorials:
$$
{}_{11}C_{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = \frac{7,\!920}{24} = 330
$$

### Summary of Results

$$
\begin{tabular}{|c|c|}
\hline
$$\mathbf{Expression}$$ & $$\mathbf{Value}$$ \
\hline
$${}_{11}P_{5}$$ & $$55,\!440$$ \
\hline
$${}_{11}C_{4}$$ & $$330$$ \
\hline
\end{tabular}
$$

**Final Answers:**
- $$ {}_{11}P_{5} = 55,\!440 $$
- $$ {}_{11}C_{4} = 330 $$
$$ \begin{tabular}{|c|c|} \hline $$\mathbf{Expression}$$ & $$\mathbf{Value}$$ \ \hline $${}_{11}P_{5}$$ & $$55,\!440$$ \ \hline $${}_{11}C_{4}$$ & $$330$$ \ \hline \end{tabular} $$ **Final Answers:** - $$ {}_{11}P_{5} = 55,\!440 $$ - $$ {}_{11}C_{4} = 330 $$

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