Evaluate $$ { }_{11} \mathrm{C}_{3} $$. $$ { }_{11} \mathrm{C}_{3}=\square $$ (Simplify your answer. Type an integer or a fraction

Question

UpStudy AI · Accepted Answer

We start by writing the combination formula:
  
$$
{ }_{n}C_{k}=\frac{n!}{k!(n-k)!}
$$

Substitute $$ n=11 $$ and $$ k=3 $$:

$$
{ }_{11}C_{3}=\frac{11!}{3!(11-3)!}=\frac{11!}{3!8!}
$$

Next, simplify by writing the factorial $$ 11! $$ as:

$$
11!=11\times10\times9\times8!
$$

This allows us to cancel the $$ 8! $$ in the numerator and denominator:

$$
{ }_{11}C_{3}=\frac{11\times10\times9\times8!}{3!\times8!}=\frac{11\times10\times9}{3!}
$$

Recall that:

$$
3! = 3\times2\times1=6
$$

Now, compute the multiplication in the numerator:

$$
11\times10\times9=990
$$

Finally, divide by $$ 6 $$:

$$
{ }_{11}C_{3}=\frac{990}{6}=165
$$

Thus, the answer is:

$$
{ }_{11}C_{3}=165
$$
$$ { }_{11}C_{3} = 165 $$

Evaluate \( { }_{11} \mathrm{C}_{3} \). \( { }_{11} \mathrm{C}_{3}=\square \) (Simplify your answer. Type an integer or a fraction