Evaluate $$ { }_{9} \mathrm{P}_{3} $$ and $$ { }_{9} C_{4} $$. $$ { }_{9} P_{3}=\square $$ $$ { }_{9} C_{4}=\square $$

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UpStudy AI · Accepted Answer

To evaluate $$ { }_{9} \mathrm{P}_{3} $$ and $$ { }_{9} C_{4} $$, we will use the formulas for permutations and combinations.

1. **Permutations**: The formula for permutations is given by:
   $$
   { }_{n} P_{r} = \frac{n!}{(n-r)!}
   $$
   For $$ { }_{9} P_{3} $$:
   $$
   { }_{9} P_{3} = \frac{9!}{(9-3)!} = \frac{9!}{6!}
   $$
   Now, we can simplify this:
   $$
   9! = 9 \times 8 \times 7 \times 6!
   $$
   Therefore,
   $$
   { }_{9} P_{3} = \frac{9 \times 8 \times 7 \times 6!}{6!} = 9 \times 8 \times 7
   $$
   Now, calculating $$ 9 \times 8 \times 7 $$:
   $$
   9 \times 8 = 72
   $$
   $$
   72 \times 7 = 504
   $$
   Thus,
   $$
   { }_{9} P_{3} = 504
   $$

2. **Combinations**: The formula for combinations is given by:
   $$
   { }_{n} C_{r} = \frac{n!}{r!(n-r)!}
   $$
   For $$ { }_{9} C_{4} $$:
   $$
   { }_{9} C_{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!}
   $$
   We can simplify this as follows:
   $$
   9! = 9 \times 8 \times 7 \times 6 \times 5!
   $$
   Therefore,
   $$
   { }_{9} C_{4} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{4! \times 5!} = \frac{9 \times 8 \times 7 \times 6}{4!}
   $$
   Now, calculating $$ 4! $$:
   $$
   4! = 4 \times 3 \times 2 \times 1 = 24
   $$
   Now we can calculate $$ 9 \times 8 \times 7 \times 6 $$:
   $$
   9 \times 8 = 72
   $$
   $$
   72 \times 7 = 504
   $$
   $$
   504 \times 6 = 3024
   $$
   Now, we divide by $$ 24 $$:
   $$
   { }_{9} C_{4} = \frac{3024}{24} = 126
   $$

Finally, we have:
$$
{ }_{9} P_{3} = 504
$$
$$
{ }_{9} C_{4} = 126
$$

Thus, the answers are:
$$
{ }_{9} P_{3} = 504
$$
$$
{ }_{9} C_{4} = 126
$$
$$ { }_{9} P_{3} = 504 $$ $$ { }_{9} C_{4} = 126 $$

Evaluate and .

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