$$ 1 \leftarrow \quad \begin{array}{l} ext { Evaluate the expressi } \ ext { a) }{ }_{12} \mathrm{C}_{0} \ ext { b) }{ }_{12} \mathrm{P}_{0}\end{array} $$

Question

$$ 1 \leftarrow \quad \begin{array}{l}	ext { Evaluate the expressi } \ 	ext { a) }{ }_{12} \mathrm{C}_{0} \ 	ext { b) }{ }_{12} \mathrm{P}_{0}\end{array} $$

UpStudy AI · Accepted Answer

We need to evaluate the following expressions:
a) $$ _{12}C_{0} $$ (combinations)
b) $$ _{12}P_{0} $$ (permutations)
For combinations, the formula is given by:
$$
_{n}C_{r} = \frac{n!}{r!(n-r)!}
$$
Substituting $$ n = 12 $$ and $$ r = 0 $$:
$$
_{12}C_{0} = \frac{12!}{0!(12-0)!} = \frac{12!}{0! \cdot 12!}
$$
For permutations, the formula is given by:
$$
_{n}P_{r} = \frac{n!}{(n-r)!}
$$
Substituting $$ n = 12 $$ and $$ r = 0 $$:
$$
_{12}P_{0} = \frac{12!}{(12-0)!} = \frac{12!}{12!}
$$
Now, we can evaluate both expressions:
1. $$ _{12}C_{0} = \frac{12!}{0! \cdot 12!} = 1 $$
2. $$ _{12}P_{0} = \frac{12!}{12!} = 1 $$
The results are: a) $$ _{12}C_{0} = 1 $$ b) $$ _{12}P_{0} = 1 $$

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